%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.02", %%% date = "18 May 2021", %%% time = "16:35:02 MDT", %%% filename = "ijnt.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% FAX = "+1 801 581 4148", %%% URL = "http://www.math.utah.edu/~beebe", %%% checksum = "64875 54117 286583 2659059", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "BibTeX; bibliography; International Journal %%% of Number Theory (IJNT)", %%% license = "public domain", %%% supported = "yes", %%% docstring = "This is a bibliography of the International %%% Journal of Number Theory (IJFCS) (CODEN %%% none, ISSN 1793-0421 (print), 1793-7310 %%% (electronic)), published by World %%% Scientific. %%% %%% Publication began with volume 1, number 1, in %%% March 2005, and the number of issues per %%% volume has increased from 4 (2005--2007) to 6 %%% (2008) to 8 (2009--2015) to 10 (2016--). %%% %%% The journal has World-Wide Web site at %%% %%% http://ejournals.wspc.com.sg/ijnt %%% https://www.worldscientific.com/worldscinet/ijnt %%% %%% At version 1.02, the COMPLETE year coverage %%% looked like this: %%% %%% 2005 ( 37) 2011 ( 124) 2017 ( 154) %%% 2006 ( 39) 2012 ( 119) 2018 ( 168) %%% 2007 ( 39) 2013 ( 122) 2019 ( 132) %%% 2008 ( 72) 2014 ( 125) 2020 ( 119) %%% 2009 ( 89) 2015 ( 143) 2021 ( 46) %%% 2010 ( 111) 2016 ( 143) %%% %%% Article: 1782 %%% %%% Total entries: 1782 %%% %%% Data for the bibliography has been collected %%% primarily from the journal Web site, with %%% additional data entries from BibNet Project %%% and TeX User Group bibliography archives. %%% %%% Numerous errors in the sources noted above %%% have been corrected. Spelling has been %%% verified with the UNIX spell and GNU ispell %%% programs using the exception dictionary %%% stored in the companion file with extension %%% .sok. %%% %%% Abstract data in this file are rough, %%% sometimes truncated, and unlikely to be %%% typesettable by TeX, due to irregular markup %%% and TeXnical conversion difficulties, and to %%% the use of low-resolution bitmap images for %%% some mathematical displays. %%% %%% Title casing is, regrettably, a mixture of %%% updowncase and downcase style; most author %%% and title data at the publisher Web site are %%% uppercase, often losing critical information. %%% %%% About one percent of the articles in this %%% journal are in French; English translations %%% of titles are provided for them. %%% %%% BibTeX citation tags are uniformly chosen %%% as name:year:abbrev, where name is the %%% family name of the first author or editor, %%% year is a 4-digit number, and abbrev is a %%% 3-letter condensation of important title %%% words. Citation tags were automatically %%% generated by software developed for the %%% BibNet Project. %%% %%% In this bibliography, entries are sorted in %%% publication order, using ``bibsort -byvolume''. %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ====================================================================

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%%% ==================================================================== %%% Acknowledgement abbreviations:

@String{ack-nhfb= "Nelson H. F. Beebe, University of Utah, Department of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1 801 581 4148, e-mail: \path|beebe@math.utah.edu|, \path|beebe@acm.org|, \path|beebe@computer.org| (Internet), URL: \path|http://www.math.utah.edu/~beebe/|"}

%%% ==================================================================== %%% Journal abbreviations:

@String{j-INT-J-NUMBER-THEORY= "International Journal of Number Theory (IJNT)"}

%%% ==================================================================== %%% Bibliography entries:

@Article{Bourgain:2005:MSP, author = "J. Bourgain", title = "More on the Sum--Product Phenomenon in Prime Fields and Its Applications", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "1--32", month = mar, year = "2005", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042105000108", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000108", abstract = "In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involved. It is shown that if and p$^{\varepsilon }$ < |B|, |C| < |A| < p$^{1 - \varepsilon }$, then |A + B| + |A \cdotp C| > p$^{\delta (\varepsilon)}$ |A|. Next we exploit the Szemer{\'e}di--Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from to satisfies |F(A,B)| > p$^{\beta }$ for some \beta = \beta (\alpha) > \alpha whenever and $ |A| \sim |B| \sim p^\alpha $, $ 0 < \alpha < 1$. The exponential sum $ \sum_{x \in A, y \in B}$ \varepsilon$_p$ (axy+bx$^2$ y$^2$), ab \neq 0 (mod p), may be estimated nontrivially for arbitrary sets satisfying |A|, |B| > p$^{\rho }$ where \rho < 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last \rho. No such examples when \rho < 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3]. Finally it is shown that if p$^{\varepsilon }$ < |A| < p$^{1 - \varepsilon }$, then always |A + A|+|A$^{-1}$ + A$^{-1}$ | > p$^{\delta (\varepsilon)}$ |A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution (mod p) of certain rational functions on the primes (cf. [12]).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2005:EFQ, author = "Heng Huat Chan and Zhi-Guo Liu and Say Tiong Ng", title = "Elliptic Functions and the Quintuple, {Hirschhorn} and {Winquist} Product Identities", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "33--43", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000017", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000017", abstract = "In this article, we derive the quintuple, Hirschhorn and Winquist product identities using the theory of elliptic functions. Our method can be used to establish generalizations of these identities due to the second author.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alkan:2005:NRT, author = "Emre Alkan and Alexandru Zaharescu", title = "Nonvanishing of the {Ramanujan} {Tau} Function in Short Intervals", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "45--51", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000029", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000029", abstract = "We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chen:2005:SEG, author = "Sin-Da Chen and Sen-Shan Huang", title = "On the series expansion of the {G{\"o}llnitz--Gordon} continued fraction", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "53--63", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000030", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000030", abstract = "We give combinatorial interpretations of the coefficients in the series expansions of the G{\"o}llnitz--Gordon continued fraction and its reciprocal. These combinatorial results enable us to determine the signs of the coefficients. At the end, we also derive some interesting identities involving the coefficients.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ivic:2005:MTS, author = "Aleksandar Ivi{\'c}", title = "The {Mellin} Transform of the Square of {Riemann}'s Zeta-Function", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "65--73", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000042", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000042", abstract = "Let. A result concerning analytic continuation of $ Z_1 $ (s) to {\mathbb{C}} is proved, and also a result relating the order of to the order of.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ono:2005:APC, author = "Ken Ono and Yuichiro Taguchi", title = "$2$-Adic Properties of Certain Modular Forms and Their Applications to Arithmetic Functions", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "75--101", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000066", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000066", abstract = "It is a classical observation of Serre that the Hecke algebra acts locally nilpotently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular {$L$}-functions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jenkins:2005:APT, author = "Paul Jenkins", title = "$p$-adic properties for traces of singular moduli", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "103--107", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S179304210500011X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210500011X", abstract = "We examine the $p$-adic properties of Zagier's traces $ \Tr (d)$ of the singular moduli of discriminant $ - d$. In a recent preprint, Edixhoven proved that if $p$ is prime and $ \frac {-d}{p} = 1$, then $ \Tr (p^{2n} d) \equiv 0 (\bmod p^n)$. We compute an exact formula for $ \Tr (p^{2n}d)$ which immediately gives Edixhoven's result as a corollary.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kedlaya:2005:LMA, author = "Kiran S. Kedlaya", title = "Local monodromy of $p$-adic differential equations: an overview", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "109--154", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S179304210500008X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210500008X", abstract = "This primarily expository article collects together some facts from the literature about the monodromy of differential equations on a $p$-adic (rigid analytic) annulus, though often with simpler proofs. These include Matsuda's classification of quasi-unipotent \nabla -modules, the Christol--Mebkhout construction of the ramification filtration, and the Christol--Dwork Frobenius antecedent theorem. We also briefly discuss the $p$-adic local monodromy theorem without proof.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Brueggeman:2005:NCN, author = "Sharon Brueggeman", title = "The Nonexistence of Certain Nonsolvable {Galois} Extensions of Number Fields of Small Degree", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "1", pages = "155--160", month = mar, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000121", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000121", abstract = "Serre's conjecture predicts the nonexistence of certain nonsolvable Galois extensions of {$ \mathbb {Q} $} which are unramified outside one small prime. These nonexistence theorems have been proven by the techniques of discriminant bounding. In this paper, we will apply these techniques to nonsolvable extensions of small degree number fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Choi:2005:STS, author = "S. K. K. Choi and A. V. Kumchev and R. Osburn", title = "On Sums of Three Squares", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "161--173", month = jun, year = "2005", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042105000054", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000054", abstract = "Let r$_3$ (n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two alternative proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of r$_3$ (n).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Adiga:2005:GRB, author = "Chandrashekar Adiga and Shaun Cooper and Jung Hun Han", title = "A General Relation Between Sums of Squares and Sums of Triangular Numbers", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "175--182", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000078", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000078", abstract = "Let r$_k$ (n) and t$_k$ (n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result r$_k$ (8n + k) = c$_k$ t$_k$ (n), which holds for 1 \leq k \leq 7, where c$_k$ is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Maier:2005:CGE, author = "H. Maier and A. Sankaranarayanan", title = "On a Certain General Exponential Sum", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "183--192", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000224", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000224", abstract = "In this paper we study the general exponential sum related to multiplicative functions $ f(n) $ with $ |f(n)| \leq 1 $, namely we study the sum $ F(x, \alpha) = \sum_{n \leq x} f(n) e(n \alpha) $ and obtain a non-trivial upper bound when $ \alpha $ is a certain type of rational number.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Williams:2005:CS, author = "Kenneth S. Williams", title = "The Convolution Sum $ \sum_{m < n / 9} \sigma (m) \sigma (n - 9 m) $", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "193--205", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000091", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000091", abstract = "The evaluation of the sum $ \sum_{m < n / 9} \sigma (m) \sigma (n - 9 m) $ is carried out for all positive integers $n$. This evaluation is used to detemine the number of solutions to $ n = x_1^2 + x_1 x_2 + x_2^2 + x_3^2 + x_3 x_4 + x_4^2 + 3 (x_5^2 + x_5 x_6 + x_6^2 + x_7^2 + x_7 x_8 + x_8^2)$ in integers $ x_1$, $ x_2$, $ x_3$, $ x_4$, $ x_5$, $ x_6$, $ x_7$, $ x_8$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2005:HMP, author = "Tsz Ho Chan", title = "Higher Moments of Primes in Short Intervals {II}", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "207--214", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000169", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000169", abstract = "Given good knowledge on the even moments, we derive asymptotic formulas for \lambda th moments of primes in short intervals and prove ``equivalence'' result on odd moments. We also provide numerical evidence in support of these results.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lovejoy:2005:TSC, author = "Jeremy Lovejoy", title = "A Theorem on Seven-Colored Overpartitions and Its Applications", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "215--224", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000157", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000157", abstract = "A $q$-series identity in four parameters is established and interpreted as a statement about 7-colored overpartitions. As corollaries some overpartition theorems of the Rogers--Ramanujan type and some weighted overpartition theorems are exhibited. Among these are overpartition analogues of classical partition theorems of Schur and G{\"o}llnitz.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Eie:2005:EGE, author = "Minking Eie and Wen-Chin Liaw and Fu-Yao Yang", title = "On Evaluation of Generalized {Euler} Sums of Even Weight", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "225--242", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000182", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000182", abstract = "The classical Euler sum $ S_{p, q} = \sum_{k = 1}^\infty \frac {1}{k^q} \sum_{j = 1}^k \frac {1}{j^p} $ cannot be evaluated when the weight $ p + q $ is even unless $ p = 1 $ or $ p = q $ or $ (p, q) = (2, 4) $ or $ (p, q) = (4, 2) $ [7]. However it is a different story if instead we consider the alternating sums $ G_{p, q}^{-, -} = \sum_{k = 0}^\infty \frac {( - 1)^k}{(2 k + 1)^q} \sum_{j = 1}^k \frac {( - 1)^{j + 1}}{j^p} $ and $ G_{p, q}^{+, -} = \sum_{k = 0}^\infty \frac {( - 1)^k}{(2 k + 1)^q} \sum_{j = 1}^k \frac {1}{j^p} $. They can be evaluated for even weight $ p + q $. In this paper, we shall evaluate a family of generalized Euler sums containing $ G_{p, q}^{-, -} $ when the weight $ p + q $ is even via integral transforms of Bernoulli identities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Delaunay:2005:MOT, author = "Christophe Delaunay", title = "Moments of the Orders of {Tate--Shafarevich} Groups", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "243--264", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000133", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000133", abstract = "We give some conjectures for the moments of the orders of the Tate--Shafarevich groups of elliptic curves belonging to a family of quadratic twists. These conjectures follow from the predictions on {$L$}-functions given by the random matrix theory [12,5] and from the Birch and Swinnerton--Dyer conjecture. Furthermore, including the Cohen--Lenstra type heuristics for Tate--Shafarevich groups, we obtain some conjectural estimates for the regulator of rank 1 elliptic curves in a family of quadratic twists.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baier:2005:LSI, author = "Stephan Baier and Liangyi Zhao", title = "Large Sieve Inequality with Characters for Powerful Moduli", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "265--279", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000170", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000170", abstract = "In this paper we aim to generalize the results in [1, 2, 19] and develop a general formula for large sieve with characters to powerful moduli that will be an improvement to the result in [19].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Petsche:2005:QVB, author = "Clayton Petsche", title = "A Quantitative Version of {Bilu}'s Equidistribution Theorem", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "281--291", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000145", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000145", abstract = "We use Fourier-analytic methods to give a new proof of Bilu's theorem on the complex equidistribution of small points on the one-dimensional algebraic torus. Our approach yields a quantitative bound on the error term in terms of the height and the degree.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baoulina:2005:PC, author = "Ioulia Baoulina", title = "On a Problem of {Carlitz}", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "2", pages = "293--307", month = jun, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000194", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:12 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000194", abstract = "Let $ N_q $ be the number of solutions to the equation $ (x_1 + \cdots + x_n)^2 = a x_1 \ldots {} x_n $ over the finite field $ \mathbb {F}_q = \mathbb {F}_p $. Carlitz found formulas for $ N_q $ when $ n = 3 $ or $4$. In an earlier paper, we found formulas for $ N_q$ when $ d = \gcd (n 2, q - 1) = 1$ or $2$ or $3$ or $4$; and when there exists an $l$ such that $ p^l - 1 (\bmod d)$. In another paper the cases $ d = 7$ or $ 14$, $ p 2$ or $4$ $ (\bmod 7)$ were considered. In this paper, we find formulas for $ N_q$ when $ d = 8$. We also simplify formulas for $ N_q$ when $ d = 4$, $ p 1$ $ (\bmod 4)$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bugeaud:2005:PPL, author = "Yann Bugeaud and Florian Luca and Maurice Mignotte and Samir Siksek", title = "On Perfect Powers in {Lucas} Sequences", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "3", pages = "309--332", month = sep, year = "2005", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042105000236", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000236", abstract = "Let (u$_n$)$_{n \geq 0}$ be the binary recurrence sequence of integers given by u$_0$ = 0, u$_1$ = 1 and u$_{n + 2}$ = 2(u$_{n + 1}$ + u$_n$). We show that the only positive perfect powers in this sequence are u$_1$ = 1 and u$_4$ = 16. We further discuss the problem of determining perfect powers in Lucas sequences in general.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Beck:2005:DAG, author = "Matthias Beck and Bruce C. Berndt and O-Yeat Chan and Alexandru Zaharescu", title = "Determinations of Analogues of {Gauss} Sums and Other Trigonometric Sums", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "3", pages = "333--356", month = sep, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000200", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000200", abstract = "Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated with primitive odd characters. In this paper, we establish two general theorems involving both sines and cosines, with more attention given to cosine sums in the several examples that we provide.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Roy:2005:SAC, author = "Damien Roy", title = "Simultaneous Approximation by Conjugate Algebraic Numbers in Fields of Transcendence Degree One", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "3", pages = "357--382", month = sep, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000212", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000212", abstract = "We present a general result of simultaneous approximation to several transcendental real, complex or $p$-adic numbers \xi$_1$, \ldots, \xi$_t$ by conjugate algebraic numbers of bounded degree over {$ \mathbb {Q}$}, provided that the given transcendental numbers \xi$_1$, \ldots, \xi$_t$ generate over {$ \mathbb {Q}$} a field of transcendence degree one. We provide sharper estimates for example when \xi$_1$, \ldots, \xi$_t$ form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at \xi$_1$, \ldots, \xi$_t$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alkan:2005:AFS, author = "Emre Alkan and Alexandru Zaharescu and Mohammad Zaki", title = "Arithmetical Functions in Several Variables", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "3", pages = "383--399", month = sep, year = "2005", DOI = "https://doi.org/10.1142/S179304210500025X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210500025X", abstract = "In this paper we investigate the ring A$_r$ (R) of arithmetical functions in r variables over an integral domain R. We study a class of absolute values, and a class of derivations on A$_r$ (R). We show that a certain extension of A$_r$ (R) is a discrete valuation ring. We also investigate the metric structure of the ring A$_r$ (R).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Suzuki:2005:RBZ, author = "Masatoshi Suzuki", title = "A Relation Between the Zeros of Two Different {$L$}-Functions Which Have an {Euler} Product and Functional Equation", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "3", pages = "401--429", month = sep, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000248", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000248", abstract = "As automorphic {$L$}-functions or Artin {$L$}-functions, several classes of {$L$}-functions have Euler products and functional equations. In this paper we study the zeros of {$L$}-functions which have Euler products and functional equations. We show that there exists a relation between the zeros of the Riemann zeta-function and the zeros of such {$L$}-functions. As a special case of our results, we find relations between the zeros of the Riemann zeta-function and the zeros of automorphic {$L$}-functions attached to elliptic modular forms or the zeros of Rankin--Selberg {$L$}-functions attached to two elliptic modular forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Niederreiter:2005:ESD, author = "Harald Niederreiter and Arne Winterhof", title = "Exponential sums and the distribution of inversive congruential pseudorandom numbers with power of two modulus", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "3", pages = "431--438", month = sep, year = "2005", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042105000261", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", MRclass = "11K38 (11K45 11L07)", MRnumber = "2175100 (2006f:11092)", MRreviewer = "Igor E. Shparlinski", bibdate = "Thu Dec 22 06:50:44 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000261", abstract = "Niederreiter and Shparlinski obtained a nontrivial discrepancy bound for sequences of inversive congruential pseudorandom numbers with odd prime-power modulus. Because of technical difficulties they had to leave open the case of greatest practical interest, namely where the modulus is a power of 2. In the present paper we successfully treat this case by using recent advances in the theory of exponential sums.", acknowledgement = ack-nhfb, ajournal = "Int. J. Number Theory", fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Eie:2005:EDE, author = "Minking Eie and Yao Lin Ong and Fu Yao Yang", title = "Evaluating Double {Euler} Sums Over Rationally Deformed Simplices", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "3", pages = "439--458", month = sep, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000273", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000273", abstract = "As a natural generalization of the classical Euler sum defined by $ S_{p, q} = \sum_{k = 1}^\infty \frac {1}{k^q} \sum_{j = 1}^k \frac {1}{j^p} $, we change the upper limit of the inner summation into $ k r $, a fixed rational multiple of $k$, and obtain countable families of new sums which we call the extended Euler sums. We shall develop a systematic new method to evaluate these extended Euler sums as well as corresponding alternating sums in terms of values at non-negative integers of cosine and sine parts of the periodic zeta function when the weight $ p + q$ is odd.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Friedlander:2005:IS, author = "J. B. Friedlander and H. Iwaniec", title = "The Illusory Sieve", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "459--494", month = dec, year = "2005", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042105000303", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000303", abstract = "We study some of the extremely strong statements that can be made about the distribution of primes assuming the (unlikely) existence of exceptional Dirichlet characters. We treat this in general and then apply the results to the particular cases of primes of the form $ a^2 + b^6 $ and of elliptic curves having prime discriminant.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Edixhoven:2005:AGT, author = "Bas Edixhoven", title = "On the $p$-adic geometry of traces of singular moduli", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "495--497", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000327", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000327", abstract = "The aim of this article is to show that $p$-adic geometry of modular curves is useful in the study of $p$-adic properties of {\em traces\/} of singular moduli. In order to do so, we partly answer a question by Ono [7, Problem 7.30]. As our goal is just to illustrate how $p$-adic geometry can be used in this context, we focus on a relatively simple case, in the hope that others will try to obtain the strongest and most general results. For example, for p = 2, a result stronger than Theorem 2 is proved in [2], and a result on some modular curves of genus zero can be found in [8]. It should be easy to apply our method, because of its local nature, to modular curves of arbitrary level, as well as to Shimura curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Litsyn:2005:IFS, author = "Simon Litsyn and Vladimir Shevelev", title = "Irrational factors satisfying the little {Fermat} theorem", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "499--512", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000339", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000339", abstract = "We study possible generalizations of the little Fermat theorem when the base of the exponentiation is allowed to be a non-integer. Such bases we call Fermat factors. We attempt classification of Fermat factors, and suggest several constructions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dummigan:2005:RTO, author = "Neil Dummigan", title = "Rational Torsion on Optimal Curves", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "513--531", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000340", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000340", abstract = "Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order \ell on optimal elliptic curves over {$ \mathbb {Q} $}. When the conductor N is squarefree, \ell \nmid N and the local root number w$_p$ = -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X$_0$ (N). We also prove some results linking the vanishing of L(E,1) with the divisibility by \ell of the modular parametrization degree, fitting well with the Bloch--Kato conjecture for L(Sym$^2$ E,2), and with an earlier construction of elements in Shafarevich--Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on \ell -torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch--Kato conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Long:2005:SPM, author = "Ling Long and Yifan Yang", title = "A Short Proof of {Milne}'s Formulas for Sums of Integer Squares", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "533--551", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000364", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000364", abstract = "We give a short proof of Milne's formulas for sums of 4n$^2$ and 4n$^2$ + 4n integer squares using the theory of modular forms. Other identities of Milne are also discussed.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mollin:2005:EAL, author = "R. A. Mollin", title = "On an Elementary Approach to the {Lebesgue--Nagell} Equation", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "553--561", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000352", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000352", abstract = "We discuss the feasibility of an elementary solution to the Diophantine equation of the form x$^2$ + D = y$^n$, where D > 1, n \geq 3 and x > 0, called the Lebesgue--Nagell equation, which has recently been solved for 1 \leq D \leq 100 in [1].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Knopfmacher:2005:SFC, author = "A. Knopfmacher and M. E. Mays", title = "A Survey of Factorization Counting Functions", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "563--581", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000315", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000315", abstract = "The general field of additive number theory considers questions concerning representations of a given positive integer n as a {\em sum\/} of other integers. In particular, {\em partitions\/} treat the sums as unordered combinatorial objects, and {\em compositions\/} treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to \pm 1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a {\em product\/} of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Luca:2005:PDL, author = "Florian Luca and Pantelimon St{\u{a}}nic{\u{a}}", title = "Prime Divisors of {Lucas} Sequences and a Conjecture of {Ska{\l}ba}", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "583--591", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000285", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000285", abstract = "In this paper, we give some heuristics suggesting that if (u$_n$)$_{n \geq 0}$ is the Lucas sequence given by u$_n$ = (a$^n$- 1)/(a - 1), where a > 1 is an integer, then \omega (u$_n$) \geq (1 + o(1))log n log log n holds for almost all positive integers n.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zhang:2005:EET, author = "Liang-Cheng Zhang", title = "Explicit Evaluations of Two {Ramanujan--Selberg} Continued Fractions", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "593--601", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000297", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000297", abstract = "This paper gives explicit evaluations for two Ramanujan--Selberg continued fractions in terms of class invariants and singular moduli.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Anonymous:2005:AIV, author = "Anonymous", title = "Author Index (Volume 1)", journal = j-INT-J-NUMBER-THEORY, volume = "1", number = "4", pages = "603--605", month = dec, year = "2005", DOI = "https://doi.org/10.1142/S1793042105000376", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:13 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042105000376", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kohnen:2006:TSA, author = "Winfried Kohnen and Riccardo Salvati Manni", title = "On the Theta Series Attached to {$ D_m^+ $}-Lattices", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "1--5", month = mar, year = "2006", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042106000449", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000449", abstract = "We show that the theta series attached to the -lattice for any positive integer divisible by 8 can be explicitly expressed as a finite rational linear combination of products of two Eisenstein series.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cohen:2006:PRQ, author = "Joseph Cohen", title = "Primitive Roots in Quadratic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "7--23", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000425", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000425", abstract = "We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin's conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vulakh:2006:DA, author = "L. Ya. Vulakh", title = "{Diophantine} approximation in {$ Q(\sqrt {-5}) $} and {$ Q(\sqrt {-5}) $}", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "25--48", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000462", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000462", abstract = "The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given. Farey polygons associated with the extended Bianchi groups B$_d$, d = 5, 6, are used to reduce the problem of finding the discrete part of the Markov spectrum for the group B$_d$ to the corresponding problem for one of its maximal Fuchsian subgroup. Hermitian points in the Markov spectrum of B$_d$ are introduced for any d. Let H$^3$ be the upper half-space model of the three-dimensional hyperbolic space. If \nu is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H$^3$ with diameter 1/\nu, which depends on one continuous parameter. This phenomenon does not take place in the hyperbolic plane.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Thong:2006:CFG, author = "Nguyen Quang Do Thong", title = "Sur la conjecture faible de {Greenberg} dans le cas ab{\'e}lien $p$-d{\'e}compos{\'e}. ({French}) [{On} the weak conjecture of {Greenberg} in the abelian $p$-decomposed case]", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "49--64", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000395", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000395", abstract = "Let p be an odd prime. For any CM number field K containing a primitive pth root of unity, class field theory and Kummer theory put together yield the well known reflection inequality \lambda$^+$ \leq \lambda$^-$ between the ``plus'' and ``minus'' parts of the \lambda -invariant of K. Greenberg's classical conjecture predicts the vanishing of \lambda$^+$. We propose a weak form of this conjecture: \lambda$^+$ = \lambda$^-$ if and only if \lambda$^+$ = \lambda$^-$ = 0, and we prove it when K$^+$ is abelian, p is totally split in K$^+$, and certain conditions on the cohomology of circular units are satisfied (e.g. in the semi-simple case).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Borwein:2006:TTG, author = "Jonathan M. Borwein and David M. Bradley", title = "Thirty-two {Goldbach} variations", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "65--103", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000383", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", MRclass = "11M41 (11M06)", MRnumber = "2217795", MRreviewer = "F. Beukers", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "http://docserver.carma.newcastle.edu.au/301/; https://www.worldscientific.com/doi/10.1142/S1793042106000383", abstract = "We give thirty-two diverse proofs of a small mathematical gem --- the fundamental Euler sum identity $ \zeta (2, 1) = \zeta (3) = 8 \zeta (\bar {2}, 1) $. We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating both the wide variety of techniques fruitfully used to study such sums and the attraction of their study.", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", researcherid-numbers = "Borwein, Jonathan/A-6082-2009", unique-id = "Borwein:2006:TTG", } @Article{Chan:2006:NPS, author = "Tsz Ho Chan", title = "A Note on Primes in Short Intervals", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "105--110", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000437", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000437", abstract = "Montgomery and Soundararajan obtained evidence for the Gaussian distribution of primes in short intervals assuming a quantitative Hardy--Littlewood conjecture. In this article, we show that their methods may be modified and an average form of the Hardy--Littlewood conjecture suffices.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Takloo-Bighash:2006:RPA, author = "Ramin Takloo-Bighash", title = "A Remark on a Paper of {Ahlgren}, {Berndt}, {Yee}, and {Zaharescu}", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "111--114", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000450", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000450", abstract = "A classical theorem of Ramanujan relates an integral of Dedekind eta-function to a special value of a Dirichlet {$L$}-function at s = 2. Ahlgren, Berndt, Yee and Zaharescu have generalized this result [1]. In this paper, we generalize this result to the context of holomorphic cusp forms on the upper half space.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cooper:2006:QPI, author = "Shaun Cooper", title = "The Quintuple Product Identity", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "115--161", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000401", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000401", abstract = "The quintuple product identity was first discovered about 90 years ago. It has been published in many different forms, and at least 29 proofs have been given. We shall give a comprehensive survey of the work on the quintuple product identity, and a detailed analysis of the many proofs.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{El-Mahassni:2006:DNC, author = "Edwin D. El-Mahassni and Arne Winterhof", title = "On the Distribution of Nonlinear Congruential Pseudorandom Numbers in Residue Rings", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "1", pages = "163--168", month = mar, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000413", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000413", abstract = "The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new type of discrepancy bound for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers over a ring of integers {\mathbb{Z}}$_M$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Walling:2006:AHO, author = "Lynne H. Walling", title = "Action of {Hecke} Operators on {Siegel} Theta Series {I}", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "169--186", month = jun, year = "2006", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042106000516", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000516", abstract = "We apply the Hecke operators T(p) and to a degree n theta series attached to a rank 2k {\mathbb{Z}}-lattice L, n \leq k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the theta series under these Hecke operators can be realized as a sum of theta series attached to certain closely related lattices, thereby generalizing the Eichler Commutation Relation (similar to some work of Freitag and of Yoshida). We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We show the eigenvalue for T(p) is \in (k - n, n), and the eigenvalue for T\prime$_j$ (p$^2$) (a specific linear combination of T$_0$ (p$^2$),\ldots, T$_j$ (p$^2$)) is p$^{j(k - n) + j(j - 1) / 2}$ \beta (n,j)\in (k-j,j) where \beta (*,*), \in (*,*) are elementary functions (defined below).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bringmann:2006:BBA, author = "Kathrin Bringmann and Benjamin Kane and Winfried Kohnen", title = "On the Boundary Behavior of Automorphic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "187--194", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000565", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000565", abstract = "We investigate the boundary behavior of modular forms f on the full modular group. We first show that $ \{ x \in [0, 1] | \lim_{y \rightarrow 0^+} y^{k / 2} |f(x + i y)| \mathrm {exists} \} $ is contained in a set of Lebesgue measure 0. In particular, we recover the well-known fact that the real axis is a natural boundary of definition for f. On the other hand, using the Rankin--Selberg Dirichlet series attached to f, we show that taking the limit over the ``average'' over all x \in [0,1] behaves ``well''. Our results also apply to Maass wave forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bennett:2006:GTB, author = "Michael A. Bennett and Alain Togb{\'e} and P. G. Walsh", title = "A Generalization of a Theorem of {Bumby} on Quartic {Diophantine} Equations", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "195--206", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000474", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000474", abstract = "Bumby proved that the only positive integer solutions to the quartic Diophantine equation 3X$^4$- 2Y$^2$ = 1 are (X, Y) = (1, 1),(3, 11). In this paper, we use Thue's hypergeometric method to prove that, for each integer m \geq 1, the only positive integers solutions to the Diophantine equation (m$^2$ + m + 1)X$^4$- (m$^2$ + m)Y$^2$ = 1 are (X,Y) = (1, 1),(2m + 1, 4m$^2$ + 4m + 3).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Laishram:2006:GCC, author = "Shanta Laishram and T. N. Shorey", title = "{Grimm}'s Conjecture on Consecutive Integers", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "207--211", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000498", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000498", abstract = "For positive integers n and k, it is possible to choose primes P$_1$, P$_2$, \ldots, P$_k$ such that P$_i$ | (n + i) for 1 \leq i \leq k whenever n + 1, n + 2,\ldots, n + k are all composites and n \leq 1.9 $ \times $ 10$^{10}$. This provides a numerical verification of Grimm's Conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hirschhorn:2006:CMS, author = "Michael D. Hirschhorn", title = "The Case of the Mysterious Sevens", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "213--216", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000486", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000486", abstract = "We give a simple, direct proof of a theorem involving partitions into distinct parts, where multiples of 7 come in two colours.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bremner:2006:TRP, author = "Andrew Bremner and Richard K. Guy", title = "Triangle-Rectangle Pairs with a Common Area and a Common Perimeter", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "217--223", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000504", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000504", abstract = "We solve a problem of Bill Sands, to find pairs of Heron triangles and rectangles, such as (5,5,6) & [2 $ \times $ 6] or (13,20,21) & [6 $ \times $ 21] which have a common area and a common perimeter. The original question was posed for right-angled triangles, but there are no nondegenerate such. There are infinitely many isosceles triangles and these have been exhibited by Guy. Here we solve the general problem; the triangle-rectangle pairs are parametrized by a family of elliptic curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Tolev:2006:DTS, author = "D. I. Tolev", title = "On the distribution of $r$-tuples of squarefree numbers in short intervals", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "225--234", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S179304210600053X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210600053X", abstract = "We consider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also establish an asymptotic formula if the interval is not very short.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Spearman:2006:DCC, author = "Blair K. Spearman and Kenneth S. Williams", title = "On the Distribution of Cyclic Cubic Fields with Index $2$", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "235--247", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000541", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000541", abstract = "In this paper we prove an analogue of Mertens' theorem for primes of each of the forms a$^2$ +27b$^2$ and 4a$^2$ +2ab+7b$^2$ and then use this result to determine an asymptotic formula for the number of positive integers n \leq x which are discriminants of cyclic cubic fields with each such field having field index 2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Grekos:2006:VTC, author = "G. Grekos and L. Haddad and C. Helou and J. Pihko", title = "Variations on a Theme of {Cassels} for Additive Bases", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "249--265", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000553", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000553", abstract = "We introduce the notion of caliber, cal(A, B), of a strictly increasing sequence of natural numbers A with respect to another one B, as the limit inferior of the ratio of the nth term of A to that of B. We further consider the limit superior t(A) of the average order of the number of representations of an integer as a sum of two elements of A. We give some basic properties of each notion and we relate the two together, thus yielding a generalization, of the form t(A) \leq t(B)/cal(A, B), of a result of Cassels specific to the case where A is an additive basis of the natural numbers and B is the sequence of perfect squares. We also provide some formulas for the computation of t(A) in a large class of cases, and give some examples.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kowalski:2006:RQT, author = "E. Kowalski", title = "On the Rank of Quadratic Twists of Elliptic Curves Over Function Fields", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "267--288", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000528", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000528", abstract = "We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/F$_q$ (C) over a function field over a finite field that have rank \geq 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gaborit:2006:ELG, author = "Philippe Gaborit and Ann Marie Natividad and Patrick Sol{\'e}", title = "{Eisenstein} Lattices, {Galois} Rings and Quaternary Codes", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "289--303", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000577", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000577", abstract = "Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) {\mathbb{Z}}$_4$-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual {\mathbb{Z}}$_4$ codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual {$ \mathbb {F} $}$_4$-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Loh:2006:ACP, author = "Po-Ru Loh and Robert C. Rhoades", title = "$p$-adic and combinatorial properties of modular form coefficients", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "2", pages = "305--328", month = jun, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000590", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000590", abstract = "For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Ap{\'e}ry numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr$_{2k}$ (\Gamma$_0$ (N), n) of the Hecke operator T$_n$ acting on the space of cusp forms S$_{2k}$ (\Gamma$_0$ (N)). We show that for (n, N) = 1, these traces interpolate $p$-adically in the weight aspect.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Flicker:2006:TCS, author = "Yuval Z. Flicker and Dmitrii Zinoviev", title = "Twisted Character of a Small Representation of {$ \mathrm {Gl}(4) $}", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "329--350", month = sep, year = "2006", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042106000589", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000589", abstract = "We compute by a purely local method the (elliptic) $ \theta $-twisted character $ \chi_{\pi Y}$ of the representation \pi_Y = I_{(3, 1)} (1_3 \times \chi_Y) of G = GL(4, F), where F is a $p$-adic field, p \neq 2, and Y is an unramified quadratic extension of F; \chi_Y is the nontrivial character of F^{\times} /N_{Y/F} Y^{\times}. The representation \pi_Y is normalizedly induced from, m_i \in GL(i, F), on the maximal parabolic subgroup of type (3, 1); \theta is the ``transpose-inverse'' involution of G. We show that the twisted character \chi_{\pi Y} of \pi_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in C_Y = C_Y (F)=``(GL(2, Y)/F^{\times})_F is minus its value at the other class within the twisted stable conjugacy class. It is 0 at the classes without norm in C_Y. Moreover \pi_Y is the endoscopic lift of the trivial representation of C_Y. We deal only with unramified Y/F, as globally this case occurs almost everywhere. The case of ramified Y/F would require another paper. Our C_Y = ``(R_{Y/F} GL(2)/GL(1))_F '' has Y-points C_Y (Y) = {(g, g\prime) \in GL(2, Y) \times GL(2, Y); det(g) = det(g\prime)}/Y^{\times} (Y^{\times} embeds diagonally); \sigma(\neq 1) in Gal(Y/F) acts by \sigma (g, g\prime) = (\sigma g\prime, \sigma g). It is a \theta -twisted elliptic endoscopic group of GL(4). Naturally this computation plays a role in the theory of lifting of C_Y and GSp(2) to GL(4) using the trace formula, to be discussed elsewhere. Our work extends --- to the context of nontrivial central characters --- the work of [7], where representations of PGL(4, F) are studied. In [7] we develop a 4-dimensional analogue of the model of the small representation of PGL(3, F) introduced by the first author and Kazhdan in [5] in a 3-dimensional case, and we extend the local method of computation introduced in [6]. As in [7] we use here the classification of twisted (stable) regular conjugacy classes in GL(4, F) of [4], motivated by Weissauer [13].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baoulina:2006:EFF, author = "Ioulia Baoulina", title = "On the Equation Over a Finite Field", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "351--363", month = sep, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000607", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000607", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hbaib:2006:BDC, author = "M. Hbaib and M. Mkaouar", title = "Sur le b{\^e}ta-d{\'e}veloppement de $1$ dans le corps des s{\'e}ries formelles. ({French}) [{On} the beta-development of $1$ in the body of formal series]", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "365--378", month = sep, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000619", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000619", abstract = "Let \beta be a fixed element of {$ \mathbb {F} $}$_q$ ((X$^{-1}$)) with polynomial part of degree \geq 1, then any formal power series can be represented in base \beta, using the transformation T$_{\beta }$: f \mapsto {\beta f} of the unit disk. Any formal power series in is expanded in this way into d$_{\beta }$ (f) = (a$_i$ (X))$_{i \geq 1}$, where. The main aim of this paper is to characterize the formal power series \beta (|\beta | > 1), such that d$_{\beta }$ (1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Lev:2006:CPA, author = "Vsevolod F. Lev", title = "Critical Pairs in {Abelian} Groups and {Kemperman}'s Structure Theorem", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "379--396", month = sep, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000620", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000620", abstract = "A well-known result by Kemperman describes the structure of those pairs (A, B) of finite subsets of an abelian group satisfying |A + B| \leq |A| + |B| -1. We establish a description which is, in a sense, dual to Kemperman's, and as an application sharpen several results due to Deshouillers, Hamidoune, Hennecart, and Plagne.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Movasati:2006:HSH, author = "H. Movasati and S. Reiter", title = "Hypergeometric Series and {Hodge} Cycles of Four Dimensional Cubic Hypersurfaces", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "397--416", month = sep, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000632", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000632", abstract = "In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four-dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of hypergeometric series on certain CM points. Our methods are based on the calculation of the Picard--Fuchs equations in higher dimensions, reducing them to the Gauss equation and then applying the Abelian Subvariety Theorem to the corresponding hypergeometric abelian varieties.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Matala-Aho:2006:VCF, author = "Tapani Matala-Aho and Ville Meril{\"a}", title = "On the values of continued fractions: $q$-series {II}", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "417--430", month = sep, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000656", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000656", abstract = "Let polynomials $ S(t) $, $ T(t) $ be given, then the convergence of the $q$-continued fraction $ T(t) + \mathbb {K}_{n = 1}^\infty \frac {S(t q^{n - 1})}{T(t q^n)}$ will be studied using the Poincar{\'e}--Perron Theorem and Frobenius series solutions of the corresponding q-difference equation $ S(t) H(q^2 t) = T(t) H(q t) + H(t)$. Our applications include a generalization of a $q$-continued fraction identity of Ramanujan and certain $q$-fractions, which arise in the theory of $q$-orthogonal polynomials.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dodson:2006:KTT, author = "M. M. Dodson and S. Kristensen", title = "{Khintchine}'s Theorem and Transference Principle for Star Bodies", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "431--453", month = sep, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000668", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000668", abstract = "Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Rodseth:2006:PPF, author = "{\O}ystein J. R{\o}dseth and James A. Sellers", title = "Partitions with Parts in a Finite Set", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "3", pages = "455--468", month = sep, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000644", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:14 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000644", abstract = "For a finite set A of positive integers, we study the partition function p$_A$ (n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for p$_A$ (n). For n in a special residue class, p$_A$ (n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of p$_A$ and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for p$_A$ (n) and its refinement.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Nicolas:2006:VIF, author = "Jean-Louis Nicolas", title = "Valeurs impaires de la fonction de partition $ p(n) $. ({French}) [{Odd} values of the partition function $ p(n) $]", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "469--487", month = dec, year = "2006", CODEN = "????", DOI = "https://doi.org/10.1142/S179304210600067X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210600067X", abstract = "Let p(n) denote the number of partitions of n, and for i = 0 (resp. 1), A$_i$ (x) denote the number of n \leq x such that p(n) is even (resp. odd). In this paper, it is proved that for some constant K > 0, holds for x large enough. This estimation slightly improves a preceding result of S. Ahlgren who obtained the above lower bound for K = 0. Let and ; the main tool is a result of J.-P. Serre about the distribution of odd values of \tau$_k$ (n). Effective lower bounds for A$_0$ (x) and A$_1$ (x) are also given.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Ayuso:2006:NST, author = "Pedro Fortuny Ayuso and Fritz Schweiger", title = "A New Symmetric Two-Dimensional Algorithm", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "489--498", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000681", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000681", abstract = "Continued fractions are deeply related to Singularity Theory, as the computation of the Puiseux exponents of a plane curve from its dual graph clearly shows. Another closely related topic is Euclid's Algorithm for computing the gcd of two integers (see [2] for a detailed overview). In the first section, we describe a subtractive algorithm for computing the gcd of n integers, related to singularities of curves in affine n-space. This gives rise to a multidimensional continued fraction algorithm whose version in dimension 2 is the main topic of the paper.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cooper:2006:RBP, author = "Joshua N. Cooper and Dennis Eichhorn and Kevin O'Bryant", title = "Reciprocals of Binary Power Series", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "499--522", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000693", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000693", abstract = "If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a + b, where a \in A and b \in B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet--Thue--Morse sequence, {0} \cup {2$^n$ :n \in \mathbb{N} }, and random sets, and we also study the distribution of densities of B for finite sets A. This problem is motivated by Euler's observation that if A is the set of n that has an odd number of partitions, then B is the set of pentagonal numbers {n(3n + 1)/2:n \in {\mathbb{Z}}}. We also elaborate the connection between this problem and the theory of de Bruijn sequences and linear shift registers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bowman:2006:CF, author = "D. Bowman and J. McLaughlin and N. J. Wyshinski", title = "A $q$-continued fraction", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "523--547", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S179304210600070X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210600070X", abstract = "We use the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for (q$^2$; q$^3$)$_{\infty }$ /(q; q$^3$)$_{\infty }$ and. In addition, we give a new proof of the famous Rogers--Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gun:2006:TZC, author = "Sanoli Gun", title = "Transcendental Zeros of Certain Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "549--553", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000711", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000711", abstract = "Kohnen showed that the zeros of the Eisenstein series E$_k$ in the standard fundamental domain other than i and \rho are transcendental. In this paper, we obtain similar results for a more general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pontreau:2006:GLB, author = "Corentin Pontreau", title = "Geometric Lower Bounds for the Normalized Height of Hypersurfaces", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "555--568", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000723", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000723", abstract = "Here we are concerned on Bogomolov's problem for hypersurfaces; we give a geometric lower bound for the height of a hypersurface of (i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic subgroup of . This is an analogue of a result of F. Amoroso and S. David who give a lower bound for the height of non-torsion hypersurfaces defined and irreducible over the rationals.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jadrijevic:2006:SRP, author = "Borka Jadrijevi{\'c} and Volker Ziegler", title = "A System of Relative {Pellian} Equations and a Related Family of Relative {Thue} Equations", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "569--590", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000735", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000735", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Byard:2006:QRD, author = "Kevin Byard", title = "On Qualified Residue Difference Sets", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "591--597", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000747", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000747", abstract = "Qualified residue difference sets of power n are known to exist for n = 2,4,6, as do similar sets that include the zero element. Both classes of sets are proved non-existent for n = 8.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kanemitsu:2006:SNT, author = "Shigeru Kanemitsu and Yoshio Tanigawa and Haruo Tsukada", title = "Some Number Theoretic Applications of a General Modular Relation", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "599--615", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000759", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000759", abstract = "We state a form of the modular relation in which the functional equation appears in the form of an expression of one Dirichlet series in terms of the other multiplied by the quotient of gamma functions and illustrate it by some concrete examples including the results of Koshlyakov, Berndt and Wigert and Bellman.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Anonymous:2006:AIV, author = "Anonymous", title = "Author Index (Volume 2)", journal = j-INT-J-NUMBER-THEORY, volume = "2", number = "4", pages = "617--619", month = dec, year = "2006", DOI = "https://doi.org/10.1142/S1793042106000760", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042106000760", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Granville:2007:PDP, author = "Andrew Granville", title = "Prime Divisors Are {Poisson} Distributed", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "1--18", month = mar, year = "2007", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042107000778", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See erratum \cite{Granville:2007:EPD}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000778", abstract = "We show that the set of prime factors of almost all integers are ``Poisson distributed'', and that this remains true (appropriately formulated) even when we restrict the number of prime factors of the integer. Our results have inspired analogous results about the distribution of cycle lengths of permutations.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Angles:2007:RI, author = "Bruno Angl{\`e}s and Thomas Herreng", title = "On a Result of {Iwasawa}", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "19--41", month = mar, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000791", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000791", abstract = "We recover a result of Iwasawa on the $p$-adic logarithm of principal units of {$ \mathbb {Q}_p(\zeta_{p^{n + 1}})$} by studying the value at s = 1 of $p$-adic {$L$}-functions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Garvan:2007:SSP, author = "Frank G. Garvan and Hamza Yesilyurt", title = "Shifted and Shiftless Partition Identities {II}", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "43--84", month = mar, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000808", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000808", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dilcher:2007:PAS, author = "Karl Dilcher and Kenneth B. Stolarsky", title = "A Polynomial Analogue to the {Stern} Sequence", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "85--103", month = mar, year = "2007", DOI = "https://doi.org/10.1142/S179304210700081X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210700081X", abstract = "We extend the Stern sequence, sometimes also called Stern's diatomic sequence, to polynomials with coefficients 0 and 1 and derive various properties, including a generating function. A simple iteration for quotients of consecutive terms of the Stern sequence, recently obtained by Moshe Newman, is extended to this polynomial sequence. Finally we establish connections with Stirling numbers and Chebyshev polynomials, extending some results of Carlitz. In the process we also obtain some new results and new proofs for the classical Stern sequence.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Choi:2007:CSP, author = "H. Timothy Choi and Ronald Evans", title = "Congruences for Sums of Powers of {Kloosterman} Sums", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "105--117", month = mar, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000821", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000821", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Shevelev:2007:D, author = "Vladimir Shevelev", title = "On divisibility of $ \binom {n - i - 1}{i - 1} $ by $i$", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "119--139", month = mar, year = "2007", DOI = "https://doi.org/10.1142/S179304210700078X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210700078X", abstract = "We investigate the function b(n) = \sum 1, where the summing is over all i for which.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hart:2007:NCM, author = "William B. Hart", title = "A New Class of Modular Equation for {Weber} Functions", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "141--157", month = mar, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000845", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000845", abstract = "We describe the construction of a new type of modular equation for Weber functions. These bear some relationship to Weber's modular equations of the {\em irrational kind}. Numerous examples of these equations are explicitly computed. We also obtain some modular equations of the irrational kind which are not present in Weber's work.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Croot:2007:SNS, author = "Ernie Croot", title = "Smooth Numbers in Short Intervals", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "1", pages = "159--169", month = mar, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000833", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000833", abstract = "We show that for any \in > 0, there exists c > 0, such that for all x sufficiently large, there are x$^{1 / 2}$ (log x)$^{-log 4 - o(1)}$ integers, all of whose prime factors are.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Wittmann:2007:PDC, author = "Christian Wittmann", title = "$l$-parts of divisor class groups of cyclic function fields of degree $l$", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "171--190", month = jun, year = "2007", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042107000857", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000857", abstract = "Let l be a prime number and K be a cyclic extension of degree l of the rational function field {$ \mathbb {F} $}$_q$ (T) over a finite field of characteristic \neq = l. Using class field theory we investigate the l-part of Pic$^0$ (K), the group of divisor classes of degree 0 of K, considered as a Galois module. In particular we give deterministic algorithms that allow the computation of the so-called (\sigma - 1)-rank and the (\sigma - 1)$^2$-rank of Pic$^0$ (K), where \sigma denotes a generator of the Galois group of K/{$ \mathbb {F} $}$_q$ (T). In the case l = 2 this yields the exact structure of the 2-torsion and the 4-torsion of Pic$^0$ (K) for a hyperelliptic function field K (and hence of the {$ \mathbb {F} $}$_q$-rational points on the Jacobian of the corresponding hyperelliptic curve over {$ \mathbb {F} $}$_q$). In addition we develop similar results for l-parts of S-class groups, where S is a finite set of places of K. In many cases we are able to prove that our algorithms run in polynomial time.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sole:2007:MFC, author = "Patrick Sol{\'e} and Dmitrii Zinoviev", title = "A {Macwilliams} Formula for Convolutional Codes", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "191--206", month = jun, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000869", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000869", abstract = "Regarding convolutional codes as polynomial analogues of arithmetic lattices, we derive a Poisson--Jacobi formula for their trivariate weight enumerator. The proof is based on harmonic analysis on locally compact abelian groups as developed in Tate's thesis to derive the functional equation of the zeta function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alkan:2007:ASG, author = "Emre Alkan", title = "Average Size of Gaps in the {Fourier} Expansion of Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "207--215", month = jun, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000870", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000870", abstract = "We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are ``finite'' on average. In particular we obtain quantitative results on the number of large values of the gap function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Amoroso:2007:MPE, author = "Francesco Amoroso", title = "Une minoration pour l'exposant du groupe des classes d'un corps engendr{\'e} par un nombre de {Salem}. ({French}) [{A} lower bound for the exponent of the group of classes of a field generated by a number of {Salem}]", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "217--229", month = jun, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000882", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000882", abstract = "In this article we extend the main result of [2] concerning lower bounds for the exponent of the class group of CM-fields. We consider a number field K generated by a Salem number \alpha. If k denotes the field fixed by \alpha \mapsto \alpha$^{-1}$ we prove, under the generalized Riemann hypothesis for the Dedekind zeta function of K, lower bounds for the relative exponent e$_{K / k}$ and the relative size h$_{K / k}$ of the class group of K with respect to the class group of k.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Royer:2007:ECS, author = "Emmanuel Royer", title = "Evaluating Convolution Sums of the Divisor Function by Quasimodular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "231--261", month = jun, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000924", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000924", abstract = "We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular forms. This extension of modular forms has been constructed by Kaneko and Zagier.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mukhopadhyay:2007:ZDE, author = "Anirban Mukhopadhyay and Kotyada Srinivas", title = "A Zero Density Estimate for the {Selberg} Class", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "263--273", month = jun, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000894", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000894", abstract = "It is well known that bounds on moments of a specific {$L$}-function can lead to zero-density result for that {$L$}-function. In this paper, we generalize this argument to all {$L$}-functions in the Selberg class by assuming a certain second power moment. As an application, it is shown that in the case of symmetric-square {$L$}-function, this result improves the existing one.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{VanWamelen:2007:NEM, author = "Paul {Van Wamelen}", title = "New Explicit Multiplicative Relations Between {Gauss} Sums", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "275--292", month = jun, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000900", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000900", abstract = "We study multiplicative identities between Gauss sums. If such an identity does not follow from the Davenport--Hasse relation and the norm relation, it is called a sign ambiguity. Until recently only a finite number of explicit sign ambiguities were known. We generalize the first infinite family of sign ambiguities as found by Murray.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sills:2007:IRR, author = "Andrew V. Sills", title = "Identities of the {Rogers--Ramanujan--Slater} Type", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "2", pages = "293--323", month = jun, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000912", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:15 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000912", abstract = "It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers--Ramanujan identities (L. J. Slater, Further identities of the Rogers--Ramanujan type, {\em Proc. London Math Soc. (2)\/} 54 (1952) 147--167) can be easily derived using just three multiparameter Bailey pairs and their associated q-difference equations. As a bonus, new Rogers--Ramanujan type identities are found along with natural combinatorial interpretations for many of these identities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Anonymous:2007:P, author = "Anonymous", title = "Preface", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "v--vi", month = sep, year = "2007", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042107001061", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001061", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Andrews:2007:FD, author = "George E. Andrews", title = "A {Fine} Dream", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "325--334", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000948", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000948", abstract = "We shall develop further N. J. Fine's theory of three parameter non-homogeneous first order q-difference equations. The object of our work is to bring the Rogers--Ramanujan identities within the purview of such a theory. In addition, we provide a number of new identities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{DeAzevedoPribitkin:2007:UPS, author = "Wladimir {De Azevedo Pribitkin}", title = "Uninhibited {Poincar{\'e}} Series", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "335--347", month = sep, year = "2007", DOI = "", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210700095X", abstract = "We introduce a class of functions that generalize the epoch-making series of Poincar{\'e} and Petersson. Our ``uninhibited Poincar{\'e} series'' permits both a complex weight and an arbitrary multiplier system that is independent of the weight. In this initial paper we provide their Fourier expansions, as well as their modular behavior. We show that they are modular integrals that possess interesting periods. Moreover, we establish with relative ease that they ``almost never'' vanish identically. Along the way we present a seemingly unknown historical truth concerning Kloosterman sums, and also an alternative approach to Petersson's factor systems. The latter depends upon a simple multiplication rule.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Berndt:2007:RCP, author = "Bruce C. Berndt", title = "{Ramanujan}'s Congruences for the Partition Function Modulo $5$, $7$, and $ 11$", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "349--354", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000961", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000961", abstract = "Using Ramanujan's differential equations for Eisenstein series and an idea from Ramanujan's unpublished manuscript on the partition function p(n) and the tau function \tau (n), we provide simple proofs of Ramanujan's congruences for p(n) modulo 5, 7, and 11.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Iwaniec:2007:FNH, author = "H. Iwaniec and W. Kohnen and J. Sengupta", title = "The First Negative {Hecke} Eigenvalue", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "355--363", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001024", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001024", abstract = "We shall improve earlier estimates on the first sign change of the Hecke eigenvalues of a normalized cuspidal newform of level N.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Goldfeld:2007:RLO, author = "Dorian Goldfeld", title = "Rank lowering operators on {$ \mathrm {GL}(n, \mathbb {R}) $}", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "365--375", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000985", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000985", abstract = "If one takes the Mellin transform of an automorphic form for GL(n) and then integrates it along the diagonal on GL(n - 1) then one obtains an automorphic form on GL(n - 1). This gives a rank lowering operator. In this paper a more general rank lowering operator is obtained by combining the Mellin transform with a sum of powers of certain fixed differential operators. The analytic continuation of the rank lowering operator is obtained by showing that the spectral expansion consists of sums of Rankin--Selberg {$L$}-functions of type GL(n) $ \times $ GL(n - 1).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mason:2007:VVM, author = "Geoffrey Mason", title = "Vector-Valued Modular Forms and Linear Differential Operators", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "377--390", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000973", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000973", abstract = "We consider holomorphic vector-valued modular forms F of integral weight k on the full modular group \Gamma = SL(2, {\mathbb{Z}}) corresponding to representations of \Gamma of arbitrary finite dimension p. Assuming that the component functions of F are linearly independent, we prove that the inequality k \geq 1 - p always holds, and that equality holds only in the trivial case when p = 1 and k = 0. For any p \geq 2, we show how to construct large numbers of representations of \Gamma for which k = 2 - p. The key idea is to consider representations of \Gamma on spaces of solutions of certain linear differential equations whose coefficients are modular forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Caulk:2007:HOH, author = "Suzanne Caulk and Lynne H. Walling", title = "{Hecke} Operators on {Hilbert--Siegel} Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "391--420", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001048", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001048", abstract = "We define Hilbert--Siegel modular forms and Hecke ``operators'' acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert--Siegel forms (i.e. with arbitrary Siegel degree) we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Schmidt:2007:CLH, author = "Thomas A. Schmidt and Mark Sheingorn", title = "Classifying Low Height Geodesics On {$ \Gamma^3 \setminus \mathcal {H} $}", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "421--438", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001012", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001012", abstract = "We show that low height-achieving non-simple geodesics on a low-index cover of the modular surface can be classified into seven types, according to the topology of highest arcs. The lowest geodesics of the signature (0;2,2,2,\infty)-orbifold are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x$^2$ + y$^2$ + z$^2$ = 3xyz, and have heights. Geodesics considered by Crisp and Moran have heights ; they conjectured that these heights, which lie in the ``mysterious region'' between 3 and the Hall ray, are isolated in the Markoff Spectrum. As a step in resolving this conjecture, we characterize the geometry on of geodesic arcs with heights strictly between 3 and 6. Of these, one type of geodesic arc cannot realize the height of any geodesic.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hassen:2007:EZF, author = "Abdul Hassen and Hieu D. Nguyen", title = "The Error Zeta Function", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "439--453", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001000", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001000", abstract = "This paper investigates a new special function referred to as the error zeta function. Derived as a fractional generalization of hypergeometric zeta functions, the error zeta function is shown to exhibit many properties analogous to its hypergeometric counterpart, including its intimate connection to Bernoulli numbers. These new properties are treated in detail and used to demonstrate a pre-functional equation satisfied by this special function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Murty:2007:OVF, author = "M. Ram Murty and V. Kumar Murty", title = "Odd Values of {Fourier} Coefficients of Certain Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "455--470", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001036", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001036", abstract = "Let f be a normalized Hecke eigenform of weight k \ge 4 on \Gamma$_0$ (N). Let \lambda$_f$ (n) denote the eigenvalue of the nth Hecke operator acting on f. We show that the number of n such that \lambda$_f$ (n) takes a given value coprime to 2, is finite. We also treat the case of levels 2$^a$ N$_0$ with a arbitrary and N$_0$ = 1, 3, 5, 15 and 17. We discuss the relationship of these results to the classical conjecture of Lang and Trotter.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Choie:2007:RBF, author = "Y. Choie and Y. Chung", title = "Representations of Binary Forms by Quaternary Forms", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "471--474", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000997", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000997", abstract = "In this paper we study a family of quaternary forms which represent almost all binary forms of a certain type. The result follows from the representation number by the genus of ternary forms and a correspondence among theta series.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Schmidt:2007:LHG, author = "Thomas A. Schmidt and Mark Sheingorn", title = "Low Height Geodesics on {$ \Gamma \setminus \mathcal {H} $}: Height Formulas and Examples", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "3", pages = "475--501", month = sep, year = "2007", DOI = "https://doi.org/10.1142/S179304210700105X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210700105X", abstract = "The Markoff spectrum of binary indefinite quadratic forms can be studied in terms of heights of geodesics on low-index covers of the modular surface. The lowest geodesics on are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x$^2$ + y$^2$ + z$^2$ = 3xyz, and have heights. Geodesics considered by Crisp and Moran have heights ; they conjectured that these heights, which lie in the ``mysterious region'' between 3 and the Hall ray, are isolated in the Markoff Spectrum. In our previous work, we classified the low height-achieving non-simple geodesics of into seven types according to the topology of highest arcs. Here, we obtain explicit formulas for the heights of geodesics of the first three types; the conjecture holds for approximation by closed geodesics of any of these types. Explicit examples show that each of the remaining types is realized.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Eliahou:2007:BMS, author = "Shalom Eliahou and Michel Kervaire", title = "Bounds on the Minimal Sumset Size Function in Groups", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "503--511", month = dec, year = "2007", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042107001085", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001085", abstract = "In this paper, we give lower and upper bounds for the minimal size \mu$_G$ (r,s) of the sumset (or product set) of two finite subsets of given cardinalities r,s in a group G. Our upper bound holds for solvable groups, our lower bound for arbitrary groups. The results are expressed in terms of variants of the numerical function \kappa$_G$ (r,s), a generalization of the Hopf--Stiefel function that, as shown in [6], exactly models \mu$_G$ (r,s) for G abelian.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Evans:2007:RRP, author = "Ronald Evans and Mark {Van Veen}", title = "Rational Representations of Primes by Binary Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "513--528", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107000936", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107000936", abstract = "Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u$^2$ + qv$^2$ has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that, where is the set of norms of algebraic integers in. Let k(q) denote the smallest positive integer k such that for all q-admissible primes p. It is shown that k(q) has subexponential but suprapolynomial growth in q, as q \rightarrow \infty.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{El-Guindy:2007:LCR, author = "Ahmad El-Guindy", title = "Linear Congruences and Relations on Spaces of Cusp Forms", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "529--539", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001097", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001097", abstract = "Let p be a prime and let f be any cusp form of level l \in {2,3,5,7,13} whose weight satisfy a certain congruence modulo (p-1). Then we exhibit explicit linear combinations of the coefficients of f that must be divisible by p. For a normalized Hecke eigenform, this translates (under mild restrictions) into the pth coefficient itself being divisible by a prime ideal above p in the ring generated by the coefficients of f. This provides many instances of the so-called non-ordinary primes. We also discuss linear relations satisfied universally on the space of modular forms of these levels. These results extend recent work of Choie, Kohnen and Ono in the level 1 case.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2007:FRR, author = "Wai Kiu Chan and A. G. Earnest and Maria Ines Icaza and Ji Young Kim", title = "Finiteness Results for Regular Definite Ternary Quadratic Forms Over", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "541--556", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001103", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001103", abstract = "Let {$ \mathfrak {o} $} be the ring of integers in a number field. An integral quadratic form over {$ \mathfrak {o} $} is called regular if it represents all integers in {$ \mathfrak {o} $} that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over {\mathbb{Z}}. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over, and thus extends the corresponding finiteness results for spinor regular quadratic forms over {\mathbb{Z}} obtained in [1,3].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pal:2007:EID, author = "Ambrus P{\'a}l", title = "On the {Eisenstein} Ideal of {Drinfeld} Modular Curves", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "557--598", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001115", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001115", abstract = "Let {$ \mathfrak {E} $}({$ \mathfrak {p} $}) denote the Eisenstein ideal in the Hecke algebra {$ \mathbb {T} $}({$ \mathfrak {p} $}) of the Drinfeld modular curve X$_0$ ({$ \mathfrak {p} $}) parameterizing Drinfeld modules of rank two over {$ \mathbb {F} $}$_q$ [T] of general characteristic with Hecke level {$ \mathfrak {p} $}-structure, where {$ \mathfrak {p} $} \triangleleft {$ \mathbb {F} $}$_q$ [T] is a non-zero prime ideal. We prove that the characteristic p of the field {$ \mathbb {F} $}$_q$ does not divide the order of the quotient {$ \mathbb {T} $}({$ \mathfrak {p} $})/{$ \mathfrak {E} $}({$ \mathfrak {p} $}) and the Eisenstein ideal {$ \mathfrak {E} $}({$ \mathfrak {p} $}) is locally principal.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Moshe:2007:CMR, author = "Yossi Moshe", title = "On a Conjecture of {McIntosh} Regarding {LP}-Sequences", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "599--610", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001139", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001139", abstract = "A sequence over {\mathbb{Z}} is an LP-sequence if for every prime p and integer n \geq 0 we have (mod p), when is a base p expansion of n. In this paper, we study sequences such that both, are LP-sequences for some d \geq 2. One of those sequences is a counter-example to a conjecture of McIntosh [15].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kraus:2007:CES, author = "Alain Kraus", title = "Courbes elliptiques semi-stables sur les corps de nombres. ({French}) [{Semi}-stable elliptical curves on number fields]", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "611--633", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001127", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001127", abstract = "Let K be a number field. In this paper, we are interested in the following problem: does there exist a constant c$_K$, which depends only on K, such that for any semi-stable elliptic curve defined over K, the Galois representation in its $p$-torsion points is irreducible whenever p is a prime number greater than c$_K$ ? In case the answer is positive, how can we get such a constant? We prove that if a certain condition is satisfied by K, the answer is positive and we obtain c$_K$ explicitly. Furthermore, we prove that this condition is realized in many situations.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Becheanu:2007:SCD, author = "Mircea Becheanu and Florian Luca and Igor E. Shparlinski", title = "On the Sums of Complementary Divisors", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "635--648", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001152", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001152", abstract = "In this paper, we study various arithmetic properties of d + n/d, where d runs through all the \tau (n) positive divisors of n. For example, denoting by \varpi (n) the number of prime values among these sums, we study how often \varpi (n) > 0 and also \varpi (n) = \tau (n), and we also evaluate the average value of \varpi (n). We estimate some character sums with d + n/d and study the distribution of quadratic nonresidues and primitive roots among these sums on average over n \leq x.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Granville:2007:EPD, author = "Andrew Granville", title = "Erratum: {``Prime Divisors Are Poisson Distributed''}", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "649--651", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001073", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See \cite{Granville:2007:PDP}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001073", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Anonymous:2007:AIV, author = "Anonymous", title = "Author Index (Volume 3)", journal = j-INT-J-NUMBER-THEORY, volume = "3", number = "4", pages = "653--654", month = dec, year = "2007", DOI = "https://doi.org/10.1142/S1793042107001164", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042107001164", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cooper:2008:CMF, author = "Yaim Cooper and Nicholas Wage and Irena Wang", title = "Congruences for Modular Forms of Non-Positive Weight", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "1--13", month = feb, year = "2008", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042108001171", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001171", abstract = "In this paper, we consider modular forms f(z) whose $q$-series expansions \sum b(n)q$^n$ have coefficients in a localized ring of algebraic integers. Extending results of Serre and Ono, we show that if f has non-positive weight, a congruence of the form b(\ell n + a) \equiv 0 (mod \nu), where \nu is a place over \ell in, can hold for only finitely many primes \ell \geq 5. To obtain this, we establish an effective bound on \ell in terms of the weight and the structure of f(z).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Takahashi:2008:APM, author = "S. Takahashi", title = "$p$-adic periods of modular elliptic curves and the level-lowering theorem", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "15--23", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001183", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001183", abstract = "An elliptic curve defined over the field of rational numbers can be considered as a complex torus. We can describe its complex periods in terms of integration of the weight-2 cusp form corresponding to the elliptic curve. In this paper, we will study an analogous description of the $p$-adic periods of the elliptic curve, considering the elliptic curve as a $p$-adic torus. An essential tool for the proof of such a description is the level-lowering theorem of Ribet, which is one of the main ingredients used in the proof of Fermat's Last Theorem.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bundschuh:2008:ARC, author = "Peter Bundschuh", title = "Arithmetical results on certain $q$-series, {I}", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "25--43", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001201", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001201", abstract = "Entire transcendental solutions of certain mth order linear q-difference equations with polynomial coefficients are considered. The aim of this paper is to give, under appropriate arithmetical conditions, lower bounds for the dimension of the K-vector space generated by 1 and the values of these solutions at m successive powers of q, where K is the rational or an imaginary quadratic number field. The main ingredients of the proofs are, first, Nesterenko's dimension estimate and its various generalizations, and secondly, Popov's method (in T{\"o}pfer's version) for the asymptotic evaluation of certain complex integrals.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Knafo:2008:ELB, author = "Emmanuel Knafo", title = "Effective Lower Bound for the Variance of Distribution of Primes in Arithmetic Progressions", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "45--56", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001213", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001213", abstract = "Through a refinement for the estimation of the effect of Siegel zeros, we show how to avoid the use of Siegel's theorem in order to obtain the first {\em effective\/} lower bound for the variance of distribution of primes in arithmetic progressions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dujella:2008:PVP, author = "Andrej Dujella and Clemens Fuchs and Florian Luca", title = "A Polynomial Variant of a Problem of {Diophantus} for Pure Powers", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "57--71", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001225", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001225", abstract = "In this paper, we prove that there does not exist a set of 11 polynomials with coefficients in a field of characteristic 0, not all constant, with the property that the product of any two distinct elements plus 1 is a perfect square. Moreover, we prove that there does not exist a set of 5 polynomials with the property that the product of any two distinct elements plus 1 is a perfect kth power with k \geq 7. Combining these results, we get an absolute upper bound for the size of a set with the property that the product of any two elements plus 1 is a pure power.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zhao:2008:WTT, author = "Jianqiang Zhao", title = "{Wolstenholme} Type Theorem for Multiple Harmonic Sums", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "73--106", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001146", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001146", abstract = "In this paper, we will study the $p$-divisibility of multiple harmonic sums (MHS) which are partial sums of multiple zeta value series. In particular, we provide some generalizations of the classical Wolstenholme's Theorem to both homogeneous and non-homogeneous sums. We make a few conjectures at the end of the paper and provide some very convincing evidence.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Rath:2008:DC, author = "P. Rath and K. Srilakshmi and R. Thangadurai", title = "On {Davenport}'s Constant", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "107--115", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001195", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001195", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kohl:2008:CCT, author = "Stefan Kohl", title = "On Conjugates of {Collatz}-Type Mappings", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "117--120", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001237", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001237", abstract = "A mapping f : {\mathbb{Z}} \rightarrow {\mathbb{Z}} is called {\em residue-class-wise affine\/} if there is a positive integer m such that it is affine on residue classes (mod m). If there is a finite set S \subset {\mathbb{Z}} which intersects nontrivially with any trajectory of f, then f is called {\em almost contracting}. Assume that f is a surjective but not injective residue-class-wise affine mapping, and that the preimage of any integer under f is finite. Then f is almost contracting if and only if there is a permutation \sigma of {\mathbb{Z}} such that f$^{\sigma }$ = \sigma$^{-1}$ \odot f \odot \sigma is either monotonically increasing or monotonically decreasing almost everywhere. In this paper it is shown that if there is no positive integer k such that applying f$^{(k)}$ decreases the absolute value of almost all integers, then \sigma cannot be residue-class-wise affine itself. The original motivation for the investigations in this paper comes from the famous 3n + 1 Conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Akbary:2008:SCP, author = "Amir Akbary and Sean Alaric and Qiang Wang", title = "On Some Classes of Permutation Polynomials", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "121--133", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001249", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001249", abstract = "Let p be a prime and q = p$^m$. We investigate permutation properties of polynomials P(x) = x$^r$ + x$^{r + s}$ + \cdots + x$^{r + ks}$ (0 < r < q - 1, 0 < s < q - 1, and k \geq 0) over a finite field {$ \mathbb {F} $}$_q$. More specifically, we construct several classes of permutation polynomials of this form over {$ \mathbb {F} $}$_q$. We also count the number of permutation polynomials in each class.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kirschenhofer:2008:FTT, author = "P. Kirschenhofer and A. Peth{\H{o}} and J. M. Thuswaldner", title = "On a Family of Three Term Nonlinear Integer Recurrences", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "135--146", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001250", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001250", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Interlando:2008:FAG, author = "J. Carmelo Interlando and Andr{\'e} Luiz Flores and Trajano Pires {Da N{\'o}brega Neto}", title = "A Family of Asymptotically Good Lattices Having a Lattice in Each Dimension", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "1", pages = "147--154", month = feb, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001262", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:16 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001262", abstract = "A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n \geq 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field {$ \mathbb {Q} $}(\zeta$_q$) where q is the smallest prime congruent to 1 modulo n.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sun:2008:LTC, author = "Zhi-Wei Sun and Daqing Wan", title = "{Lucas}-type congruences for cyclotomic $ \psi $-coefficients", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "155--170", month = apr, year = "2008", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042108001286", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001286", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kazalicki:2008:LRC, author = "Matija Kazalicki", title = "Linear Relations for Coefficients of {Drinfeld} Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "171--176", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001274", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001274", abstract = "Choie, Kohnen and Ono have recently classified the linear relations among the initial Fourier coefficients of weight k modular forms on SL$_2$ ({\mathbb{Z}}), and they employed these results to obtain particular $p$-divisibility properties of some $p$-power Fourier coefficients that are common to all modular forms of certain weights. Using this, they reproduced some famous results of Hida on non-ordinary primes. Here we generalize these results to Drinfeld modular forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Abouzaid:2008:HLA, author = "Mourad Abouzaid", title = "Heights and logarithmic $ \gcd $ on algebraic curves", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "177--197", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001298", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001298", abstract = "Let F(x,y) be an irreducible polynomial over {$ \mathbb {Q} $}, satisfying F(0,0) = 0. Skolem proved that the integral solutions of F(x,y) = 0 with fixed gcd are bounded [13] and Walsh gave an explicit bound in terms of d = gcd(x,y) and F [16]. Assuming that (0,0) is a non-singular point of the plane curve F(x,y) = 0, we extend this result to algebraic solution, and obtain an asymptotic equality instead of inequality. We show that for any algebraic solution (\alpha , \beta), the quotient h(\alpha)/log d is approximatively equal to deg$_y$ F and the quotient h(\beta)/log d to deg$_x$ F; here h(\cdotp ) is the absolute logarithmic height and d is the (properly defined) ``greatest common divisor'' of \alpha and \beta.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mortenson:2008:BDP, author = "Eric Mortenson", title = "On the Broken $1$-Diamond Partition", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "199--218", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001365", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001365", abstract = "We introduce a crank-like statistic for a different class of partitions. In [4], Andrews and Paule initiated the study of broken k-diamond partitions. Their study of the respective generating functions led to an infinite family of modular forms, about which they were able to produce interesting arithmetic theorems and conjectures for the related partition functions. Here we establish a crank-like statistic for the broken 1-diamond partition and discuss its role in congruence properties.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alaca:2008:TFI, author = "Ay{\c{s}}e Alaca and {\c{S}}aban Alaca and Mathieu F. Lemire and Kenneth S. Williams", title = "Theta Function Identities and Representations by Certain Quaternary Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "219--239", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001304", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001304", abstract = "Some new theta function identities are proved and used to determine the number of representations of a positive integer n by certain quaternary quadratic forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Harman:2008:WMV, author = "Glyn Harman", title = "{Watt}'s Mean Value Theorem and {Carmichael} Numbers", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "241--248", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001316", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001316", abstract = "It is shown that Watt's new mean value theorem on sums of character sums can be included in the method described in the author's recent work [6] to show that the number of Carmichael numbers up to x exceeds x$^{{\u {2}153}}$ for all large x. This is done by comparing the application of Watt's original version of his mean value theorem [8] to the problem of primes in short intervals [3] with the problem of finding ``small'' primes in an arithmetic progression.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Watt:2008:BMV, author = "Nigel Watt", title = "Bounds for a Mean Value of Character Sums", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "249--293", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001328", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001328", abstract = "We obtain new upper bounds for the mean squared modulus of sums $ \sum_{n \in \mathbb {N} } $ A$_n$ \chi (n), where the sequence (A$_n$) is fixed and the variable \chi belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (c$_m$) satisfying $ c_m = 0$ for $ m \notin (M / 2, M]$, and some $ \alpha (x)$ and $ \beta (y)$ (smooth functions with compact support), one has $ A_n = \sum_{u v m = n} \alpha (u) \beta (v) c_m (n \in \mathbb {N})$. There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of $P$, the largest prime factor of $q$, and improve as $ \log_q(P)$ is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Penniston:2008:ARP, author = "David Penniston", title = "Arithmetic of $ \ell $-regular partition functions", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "295--302", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001341", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001341", abstract = "Let b$_{\ell }$ (n) denote the number of \ell -regular partitions of n, where \ell is prime and 3 \leq \ell \leq 23. In this paper we prove results on the distribution of b$_{\ell }$ (n) modulo m for any odd integer m > 1 with 3 \nmid m if \ell \neq 3.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bringmann:2008:RCO, author = "Kathrin Bringmann and Jeremy Lovejoy", title = "Rank and Congruences for Overpartition Pairs", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "303--322", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001353", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001353", abstract = "The rank of an overpartition pair is a generalization of Dyson's rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of, the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Milas:2008:NTP, author = "Antun Milas and Eric Mortenson and Ken Ono", title = "Number Theoretic Properties of {Wronskians} of {Andrews--Gordon} Series", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "2", pages = "323--337", month = apr, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001377", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001377", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Nedev:2008:BSV, author = "Zhivko Nedev and Anthony Quas", title = "Balanced Sets and the Vector Game", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "339--347", month = jun, year = "2008", CODEN = "????", DOI = "https://doi.org/10.1142/S179304210800133X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210800133X", abstract = "We consider the notion of a balanced set modulo N. A nonempty set S of residues modulo N is balanced if for each x \in S, there is a d with 0 < d \leq N/2 such that x \pm d mod N both lie in S. We define \alpha (N) to be the minimum cardinality of a balanced set modulo N. This notion arises in the context of a two-player game that we introduce and has interesting connections to the prime factorization of N. We demonstrate that for p prime, \alpha (p) = \Theta (log p), giving an explicit algorithmic upper bound and a lower bound using finite field theory and show that for N composite, \alpha (N) = min$_{p|N}$ \alpha (p).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Brueggeman:2008:LCD, author = "Sharon Brueggeman and Darrin Doud", title = "Local Corrections of Discriminant Bounds and Small Degree Extensions of Quadratic Base Fields", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "349--361", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001389", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001389", abstract = "Using analytic techniques of Odlyzko and Poitou, we create tables of lower bounds for discriminants of number fields, including local corrections for ideals of known norm. Comparing the lower bounds found in these tables with upper bounds on discriminants of number fields obtained from calculations involving differents, we prove the nonexistence of a number of small degree extensions of quadratic fields having limited ramification. We note that several of our results require the locally corrected bounds.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bacher:2008:NIH, author = "Roland Bacher", title = "A New Inequality for the {Hermite} Constants", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "363--386", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001390", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001390", abstract = "We describe continuous increasing functions C$_n$ (x) such that \gamma$_n$ \geq C$_n$ (\gamma$_{n - 1}$) where \gamma$_m$ is Hermite's constant in dimension m. This inequality yields a new proof of the Minkowski--Hlawka bound \Delta$_n$ \geq \zeta (n)2$^{1 - n}$ for the maximal density \Delta$_n$ of n-dimensional lattice packings.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Coulangeon:2008:EZF, author = "Renaud Coulangeon", title = "On {Epstein}'s Zeta Function of {Humbert} Forms", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "387--401", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001407", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001407", abstract = "The Epstein \zeta function \zeta (\Gamma, s) of a lattice \Gamma is defined by a series which converges for any complex number s such that {\mathfrak{R}} s > n/2, and admits a meromorphic continuation to the complex plane, with a simple pole at s = n/2. The question as to which \Gamma, for a fixed s, minimizes \zeta (\Gamma, s), has a long history, dating back to Sobolev's work on numerical integration, and subsequent papers by Delone and Ryshkov among others. This was also investigated more recently by Sarnak and Strombergsson. The present paper is concerned with similar questions for positive definite quadratic forms over number fields, also called {\em Humbert forms}. We define Epstein zeta functions in that context and study their meromorphic continuation and functional equation, this being known in principle but somewhat hard to find in the literature. Then, we give a general criterion for a Humbert form to be {\em finally\/} \zeta {\em extreme\/}, which we apply to a family of examples in the last section.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Matsuno:2008:AII, author = "Kazuo Matsuno", title = "On the $2$-Adic {Iwasawa} Invariants of Ordinary Elliptic Curves", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "403--422", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001468", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001468", abstract = "In this paper, we give an explicit formula describing the variation of the 2-adic Iwasawa \lambda -invariants attached to the Selmer groups of elliptic curves under quadratic twists. To prove this formula, we extend some results known for odd primes p, an analogue of Kida's formula proved by Hachimori and the author and a formula given by Greenberg and Vatsal, to the case where p = 2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lau:2008:LQN, author = "Yuk-Kam Lau and Jie Wu", title = "On the Least Quadratic Non-Residue", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "423--435", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001432", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001432", abstract = "We prove that for almost all real primitive characters \chi$_d$ of modulus |d|, the least positive integer n$_{\chi d}$ at which \chi$_d$ takes a value not equal to 0 and 1 satisfies n$_{\chi d}$ \ll log|d|, and give a quite precise estimate on the size of the exceptional set.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ong:2008:EET, author = "Yao Lin Ong and Minking Eie and Wen-Chin Liaw", title = "Explicit Evaluation of Triple {Euler} Sums", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "437--451", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001420", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001420", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kochubei:2008:DCE, author = "Anatoly N. Kochubei", title = "{Dwork--Carlitz} Exponential and Overconvergence for Additive Functions in Positive Characteristic", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "453--460", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001444", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001444", abstract = "We study overconvergence phenomena for {$ \mathbb {F} $}-linear functions on a function field over a finite field {$ \mathbb {F} $}. In particular, an analog of the Dwork exponential is introduced.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Toh:2008:GTO, author = "Pee Choon Toh", title = "Generalized $m$-th order {Jacobi} theta functions and the {Macdonaldcg} identities", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "461--474", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001456", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001456", abstract = "We describe an mth order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sankaranarayanan:2008:ESC, author = "A. Sankaranarayanan and N. Saradha", title = "Estimates for the Solutions of Certain {Diophantine} Equations by {Runge}'s Method", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "475--493", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S179304210800147X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210800147X", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zhang:2008:ACS, author = "Lingrui Zhang and Qin Yue", title = "Another Case of a {Scholz}'s Theorem on Class Groups", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "495--501", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001493", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001493", abstract = "In this paper, we give necessary and sufficient conditions for 8-ranks of narrow class groups of, distinct primes p \equiv q \equiv 1 mod 4. The results are useful for numerical computations.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Fukshansky:2008:SZQ, author = "Lenny Fukshansky", title = "Small Zeros of Quadratic Forms Over", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "3", pages = "503--523", month = jun, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001481", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001481", abstract = "Let N \geq 2 be an integer, F a quadratic form in N variables over, and an $L$-dimensional subspace, 1 \leq L \leq N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over. This extends previous results of the author over number fields. All bounds on height are explicit.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baruah:2008:SSS, author = "Nayandeep Deka Baruah and Shaun Cooper and Michael Hirschhorn", title = "Sums of Squares and Sums of Triangular Numbers Induced by Partitions of $8$", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "525--538", month = aug, year = "2008", CODEN = "????", DOI = "https://doi.org/10.1142/S179304210800150X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210800150X", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chapman:2008:AWT, author = "Robin Chapman and Hao Pan", title = "$q$-analogues of {Wilson}'s theorem", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "539--547", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001511", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001511", abstract = "We give q-analogues of Wilson's theorem for the primes congruent to 1 and 3 modulo 4, respectively. Also q-analogues of two congruences due to Mordell and Chowla are established.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Schwab:2008:UFC, author = "Emil Daniel Schwab and Pentti Haukkanen", title = "A unique factorization in commutative {M{\"o}bius} monoids", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "549--561", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001523", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001523", abstract = "We show that any commutative M{\"o}bius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative M{\"o}bius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative M{\"o}bius monoid.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Oura:2008:TSR, author = "Manabu Oura and Cris Poor and David S. Yuen", title = "Towards the {Siegel} Ring in Genus Four", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "563--586", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001535", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001535", abstract = "Runge gave the ring of genus three Siegel modular forms as a quotient ring, R$_3$ /{\u{3}008}J$^{(3)}$ {\u{3}009} where R$_3$ is the genus three ring of code polynomials and J$^{(3)}$ is the difference of the weight enumerators for the e$_8$ \oplus e$_8$ and codes. Freitag and Oura gave a degree 24 relation,, of the corresponding ideal in genus four; where is also a linear combination of weight enumerators. We take another step towards the ring of Siegel modular forms in genus four. We explain new techniques for computing with Siegel modular forms and actually compute six new relations, classifying all relations through degree 32. We show that the local codimension of any irreducible component defined by these known relations is at least 3 and that the true ideal of relations in genus four is not a complete intersection. Also, we explain how to generate an infinite set of relations by symmetrizing first order theta identities and give one example in degree 32. We give the generating function of R$_5$ and use it to reprove results of Nebe [25] and Salvati Manni [37].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bonciocat:2008:CLP, author = "Nicolae Ciprian Bonciocat", title = "Congruences and {Lehmer}'s Problem", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "587--596", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001547", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001547", abstract = "We obtain explicit lower bounds for the Mahler measure for nonreciprocal polynomials with integer coefficients satisfying certain congruences.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chakraborty:2008:ECG, author = "Kalyan Chakraborty and Florian Luca and Anirban Mukhopadhyay", title = "Exponents of Class Groups of Real Quadratic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "597--611", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001559", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001559", abstract = "In this paper, we show that the number of real quadratic fields {$ \mathbb {K} $} of discriminant $ \Delta_{ \mathbb {K}} < x $ whose class group has an element of order $g$ (with $g$ even) is $ \geq x^{1 / g} / 5 $ if $ x > x_0 $, uniformly for positive integers $ g \leq (\log \log x) / (8 \log \log \log x) $. We also apply the result to find real quadratic number fields whose class numbers have many prime factors.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Masri:2008:IFF, author = "Nadia Masri", title = "Infinite Families of Formulas for Sums of Integer Squares", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "613--626", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001560", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001560", abstract = "In 2002, Milne [5, 6] obtained ten infinite families of formulas for the sums of integer squares. Recently, Long and Yang [4] reproved four of these identities using modular forms on various subgroups. In this paper, we prove the remaining six, and show that all of the identities can be proved by interpreting them in terms of modular forms for \Gamma$_0$ (4).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Clark:2008:AHP, author = "Pete L. Clark", title = "An ``anti-{Hasse} Principle'' for Prime Twists", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "627--637", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001572", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001572", abstract = "Given an algebraic curve $ C_{\mathbb {Q}} $ having points everywhere locally and endowed with a suitable involution, we show that there exists a positive density family of prime quadratic twists of C violating the Hasse principle. The result applies in particular to $ w_N$-Atkin--Lehner twists of most modular curves X$_0 (N)$ and to $ w_p$-Atkin--Lehner twists of certain Shimura curves $ X^{D+}$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pineda-Ruelas:2008:EGG, author = "Mario Pineda-Ruelas and Gabriel D. Villa-Salvador", title = "Explicit {Galois} Group Realizations", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "639--652", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001584", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001584", abstract = "We study the embedding problem with abelian kernel and we obtain a homogeneous system of equations, which leads directly to the explicit realization of a finite group with certain properties. We give an example motivated by finding explicitly nonsolitary fields of degree 18 over {$ \mathbb {Q} $}.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Farag:2008:DTR, author = "Hany M. Farag", title = "{Dirichlet} Truncations of the {Riemann} Zeta Function in the Critical Strip Possess Zeros Near Every Vertical Line", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "653--662", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001596", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001596", abstract = "We study the zeros of the finite truncations of the alternating Dirichlet series expansion of the Riemann zeta function in the critical strip. We do this with an (admittedly highly) ambitious goal in mind. Namely, that this series converges to the zeta function (up to a trivial term) in the critical strip and our hope is that if we can obtain good estimates for the zeros of these approximations it may be possible to generalize some of the results to zeta itself. This paper is a first step towards this goal. Our results show that these finite approximations have zeros near every vertical line (so no vertical strip in this region is zero-free). Furthermore, we give upper bounds for the imaginary parts of the zeros (the real parts are pinned). The bounds are numerically very large. Our tools are: the inverse mapping theorem (for a perturbative argument), the prime number theorem (for counting primes), elementary geometry (for constructing zeros of a related series), and a quantitative version of Kronecker's theorem to go from one series to another.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ash:2008:EUA, author = "Avner Ash and David Pollack", title = "Everywhere unramified automorphic cohomology for {$ \mathrm {SL}_3 (\mathbb {Z}) $}", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "663--675", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001602", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001602", abstract = "We conjecture that the only irreducible cuspidal automorphic representations for GL$_3$ /{$ \mathbb {Q}$} of cohomological type and level 1 are (up to twisting) the symmetric square lifts of classical cuspforms on GL$_2$ /{$ \mathbb {Q}$} of level 1. We present computational evidence for this conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alaca:2008:BCF, author = "Ay{\c{s}}e Alaca and {\c{S}}aban Alaca and Kenneth S. Williams", title = "{Berndt}'s Curious Formula", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "677--689", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001614", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001614", abstract = "A curious arithmetic formula deduced by Berndt from an analytic formula of Ramanujan is proved arithmetically and used to prove the formulae given by Liouville for the number of representations of a positive integer by the forms $ x^2 + y^2 + z^2 + t^2 + 2 u^2 + 2 v^2 $ and $ x^2 + y^2 + 2 z^2 + 2 t^2 + 2 u^2 + 2 v^2 $.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Boca:2008:PES, author = "Florin P. Boca", title = "A problem of {Erd{\H{o}}s}, {Sz{\"o}sz} and {Tur{\'a}n} concerning {Diophantine} approximations", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "4", pages = "691--708", month = aug, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001626", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:17 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001626", abstract = "For $ A > 0 $ and $ c > 1 $, let $ S(N, A, c) $ denote the set of those numbers $ \theta \in ]0, 1 [ $ which satisfy for some coprime integers $a$ and $b$ with $ N < b \leq c N$. The problem of the existence and computation of the limit $ f(A, c)$ of the Lebesgue measure of $ S(N, A, c)$ as $ N \to \infty $ was raised by Erd{\H{o}}s, Sz{\"u}sz and Tur{\'a}n [3]. This limit has been shown to exist by Kesten and S{\'o}s [5] using a probabilistic argument and explicitly computed when $ A c \leq 1$ by Kesten [4]. We give a complete solution proving directly the existence of this limit and identifying it in all cases.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chapman:2008:RIF, author = "Robin Chapman", title = "Representations of integers by the form $ x^2 + x y + y^2 + z^2 + z t + t^2 $", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "709--714", month = oct, year = "2008", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042108001638", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001638", abstract = "We give an elementary proof of the number of representations of an integer by the quaternary quadratic form x$^2$ + xy + y$^2$ + z$^2$ + zt + t$^2$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Languasco:2008:HLP, author = "Alessandro Languasco and Alessandro Zaccagnini", title = "On the {Hardy--Littlewood} Problem in Short Intervals", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "715--723", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S179304210800164X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210800164X", abstract = "We study the distribution of Hardy--Littlewood numbers in short intervals both unconditionally and conditionally, i.e. assuming the Riemann Hypothesis (RH). We prove that a suitable average of the asymptotic formula for the number of representations of a Hardy--Littlewood number holds in the interval [n, n + H], where H < X$^{1 - 1 / k + \in }$ and n \in [X, 2X].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kopeliovich:2008:TCI, author = "Yaacov Kopeliovich", title = "Theta Constant Identities at Periods of Coverings of Degree 3", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "725--733", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001663", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001663", abstract = "We derive relations between theta functions evaluated at period matrices of cyclic covers of order 3 ramified above 3k points.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mizuno:2008:ALS, author = "Yoshinori Mizuno", title = "A $p$-adic limit of {Siegel--Eisenstein} series of prime level $q$", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "735--746", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001729", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001729", abstract = "We show that a $p$-adic limit of a Siegel--Eisenstein series of prime level q becomes a Siegel modular form of level pq. This paper contains a simple formula for Fourier coefficients of a Siegel--Eisenstein series of degree two and prime levels.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ernvall-Hytonen:2008:ETA, author = "Anne-Maria Ernvall-Hyt{\"o}nen", title = "On the Error Term in the Approximate Functional Equation for Exponential Sums Related to Cusp Forms", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "747--756", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001730", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001730", abstract = "We give a proof for the approximate functional equation for exponential sums related to holomorphic cusp forms and derive an upper bound for the error term.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Thunder:2008:PBH, author = "Jeffrey Lin Thunder", title = "Points of Bounded Height on {Schubert} Varieties", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "757--765", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001742", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001742", abstract = "Growth estimates and asymptotic estimates are given for the number of rational points of bounded height on Schubert varieties defined over number fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hassen:2008:HBP, author = "Abdul Hassen and Hieu D. Nguyen", title = "Hypergeometric {Bernoulli} Polynomials and {Appell} Sequences", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "767--774", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001754", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001754", abstract = "There are two analytic approaches to Bernoulli polynomials B$_n$ (x): either by way of the generating function ze$^{xz}$ /(e$^z$- 1) = \sum B$_n$ (x)z$^n$ /n! or as an Appell sequence with zero mean. In this article, we discuss a generalization of Bernoulli polynomials defined by the generating function z$^N$ e$^{xz}$ /(e$^z$- T$_{N - 1}$ (z)), where T$_N$ (z) denotes the Nth Maclaurin polynomial of e$^z$, and establish an equivalent definition in terms of Appell sequences with zero moments in complete analogy to their classical counterpart. The zero-moment condition is further shown to generalize to Bernoulli polynomials generated by the confluent hypergeometric series.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Onodera:2008:BSG, author = "Kazuhiro Onodera", title = "Behavior of Some Generalized Multiple Sine Functions", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "775--796", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001651", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001651", abstract = "Our aim is to investigate the behavior of generalized multiple sine functions with general period parameters in the fundamental domain. For that, we need to calculate the number of their extremal values. By estimating their special values, we determine it in some cases including the quintuple sine function. As a consequence, we sketch their graphs.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baoulina:2008:NSE, author = "Ioulia Baoulina", title = "On the number of solutions to the equation $ (x_1 + \cdots + x_n)^2 = a x_1 \cdots x_n $ in a finite field", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "797--817", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001675", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001675", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ford:2008:CFF, author = "Kevin Ford and Igor Shparlinski", title = "On Curves Over Finite Fields with {Jacobians} of Small Exponent", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "819--826", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001687", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001687", abstract = "We show that finite fields over which there is a curve of a given genus g \geq 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of g = 1. We also show that when g = 1 or g = 2, our lower bounds on the exponent, valid for almost all finite fields {$ \mathbb {F} $}$_q$ and all curves over {$ \mathbb {F} $}$_q$, are best possible.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Leher:2008:BGN, author = "Eli Leher", title = "Bounds for the Genus of Numerical Semigroups", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "827--834", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001699", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001699", abstract = "We introduce a method to find upper and lower bounds for the genus of numerical semigroups. Using it we prove some old and new bounds for it and for the Frobenius number of the semigroup.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jarden:2008:UFR, author = "Moshe Jarden and Carlos R. Videla", title = "Undecidability of Families of Rings of Totally Real Integers", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "835--850", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001705", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001705", abstract = "Let {\mathbb{Z}}$_{tr}$ be the ring of totally real integers, Gal({$ \mathbb {Q}$}) the absolute Galois group of {$ \mathbb {Q}$}, and e a positive integer. For each \sigma = (\sigma$_1$, \ldots, \sigma$_e$) \in Gal({$ \mathbb {Q}$})$^e$ let {\mathbb{Z}}$_{tr}$ (\sigma) be the fixed ring in {\mathbb{Z}}$_{tr}$ of \sigma$_1$, \ldots, \sigma$_e$. Then, the theory of all first order sentences \theta that are true in {\mathbb{Z}}$_{tr}$ (\sigma) for almost all \sigma \in Gal({$ \mathbb {Q}$})$^e$ (in the sense of the Haar measure) is undecidable.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zieve:2008:SFP, author = "Michael E. Zieve", title = "Some Families of Permutation Polynomials Over Finite Fields", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "851--857", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001717", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001717", abstract = "We give necessary and sufficient conditions for a polynomial of the form x$^r$ (1 + x$^v$ + x$^{2v}$ + \cdots + x$^{kv}$ )$^t$ to permute the elements of the finite field {$ \mathbb {F} $}$_q$. Our results yield especially simple criteria in case (q - 1)/gcd(q - 1, v) is a small prime.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Liu:2008:PIS, author = "Yuancheng Liu", title = "On the Problem of Integer Solutions to Decomposable Form Inequalities", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "5", pages = "859--872", month = oct, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001766", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001766", abstract = "This paper proves a conjecture proposed by Chen and Ru in [1] on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F(X$_1$, \ldots, X$_m$) be a non-degenerate decomposable form with coefficients in k. We show that for every finite set of places S of k containing the archimedean places of k, for each real number \lambda < 1 and each constant c > 0, the inequality has only finitely many -non-proportional solutions, where H$_S$ (x$_1$, \ldots, x$_m$) = \Pi$_{\upsilon \in S}$ max$_{1 \leq i \leq m}$ ||x$_i$ ||$_{\upsilon }$ is the S-height.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Rosengren:2008:SSE, author = "Hjalmar Rosengren", title = "Sums of Squares from Elliptic {Pfaffians}", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "873--902", month = dec, year = "2008", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042108001778", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001778", abstract = "We give a new proof of Milne's formulas for the number of representations of an integer as a sum of 4m$^2$ and 4m(m + 1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne's formulas to Schur Q-polynomials and to correlation functions for continuous dual Hahn polynomials. We also state a new formula for 2m$^2$ squares.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Balasuriya:2008:CES, author = "Sanka Balasuriya and William D. Banks and Igor E. Shparlinski", title = "Congruences and Exponential Sums with the Sum of Aliquot Divisors Function", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "903--909", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S179304210800178X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210800178X", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kamano:2008:ABN, author = "Ken Kamano", title = "$p$-adic $q$-{Bernoulli} numbers and their denominators", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "911--925", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S179304210800181X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210800181X", abstract = "We define $p$-adic q-Bernoulli numbers by using a $p$-adic integral. These numbers have good properties similar to those of the classical Bernoulli numbers. In particular, they satisfy an analogue of the von Staudt--Clausen theorem, which includes information of denominators of $p$-adic q-Bernoulli numbers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Balandraud:2008:IMN, author = "{\'E}ric Balandraud", title = "The Isoperimetric Method in Non-{Abelian} Groups with an Application to Optimally Small Sumsets", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "927--958", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001821", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001821", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gurak:2008:PHK, author = "S. Gurak", title = "Polynomials for Hyper-{Kloosterman} Sums", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "959--972", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001808", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001808", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Luca:2008:DE, author = "Florian Luca and Alain Togb{\'e}", title = "On the {Diophantine} equation $ x^2 + 2^a \cdot 5^b = y^n $", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "973--979", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001791", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001791", abstract = "In this note, we find all the solutions of the Diophantine equation x$^2$ + 2$^a$ \cdotp 5$^b$ = y$^n$ in positive integers x, y, a, b, n with x and y coprime and n \geq 3.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Walling:2008:AHO, author = "Lynne H. Walling", title = "Action of {Hecke} Operators on {Siegel} Theta Series, {II}", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "981--1008", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001845", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001845", abstract = "We apply the Hecke operators T(p)$^2$ and (1 \leq j \leq n \leq 2k) to a degree n theta series attached to a rank 2k {\mathbb{Z}}-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of, thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average theta series, extending previous work where we had the restrictions that \chi (p) = 1 and n \leq k. We also show that for j > k when \chi (p) = 1, and for j \geq k when \chi (p) = -1, and that \theta (gen L) is an eigenform for T(p)$^2$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{El-Mahassni:2008:DCD, author = "Edwin D. El-Mahassni and Domingo Gomez", title = "On the Distribution of Counter-Dependent Nonlinear Congruential Pseudorandom Number Generators in Residue Rings", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "1009--1018", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001857", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001857", abstract = "Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper, using a technique developed by Niederreiter and Shparlinski, we present discrepancy bounds for sequences of s-tuples of successive pseudorandom numbers generated by counter-dependent generators modulo a composite M.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Khanduja:2008:TD, author = "Sudesh K. Khanduja and Munish Kumar", title = "On a Theorem of {Dedekind}", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "1019--1025", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001833", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001833", abstract = "Let K = {$ \mathbb {Q} $}(\theta) be an algebraic number field with \theta in the ring A$_K$ of algebraic integers of K and f(x) be the minimal polynomial of \theta over the field {$ \mathbb {Q}$} of rational numbers. For a rational prime p, let be the factorization of the polynomial obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over {\mathbb{Z}}/p{\mathbb{Z}}. Dedekind proved that if p does not divide [A$_K$: {\mathbb{Z}}[\theta ]], then the factorization of pA$_K$ as a product of powers of distinct prime ideals is given by, with {$ \mathfrak {p} $}$_i$ = pA$_K$ + g$_i$ (\theta)A$_K$, and residual degree. In this paper, we prove that if the factorization of a rational prime p in A$_K$ satisfies the above-mentioned three properties, then p does not divide [A$_K$ :{\mathbb{Z}}[\theta ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Garthwaite:2008:CMT, author = "Sharon Anne Garthwaite", title = "The coefficients of the $ \omega (q) $ mock theta function", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "1027--1042", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001869", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001869", abstract = "In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of these functions. Recently, Zwegers proved a relationship between mock theta functions and vector-valued modular forms, and Bringmann and Ono used the theory of Maass forms and Poincar{\'e} series to prove a conjecture of Andrews, yielding an exact formula for the coefficients of the f(q) mock theta function. Here we build upon these results, using the theory of vector-valued modular forms and Poincar{\'e} series to prove an exact formula for the coefficients of the \omega (q) mock theta function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{David:2008:PLA, author = "Sinnou David and Am{\'i}lcar Pacheco", title = "Le probl{\`e}me de {Lehmer} ab{\'e}lien pour un module de {Drinfel'd}. ({French}) [{The} {Lehmer} abelien problem for a {Drinfel'd} module]", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "1043--1067", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001870", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001870", abstract = "Let \varphi be a Drinfel'd module defined over a finite extension K of {$ \mathbb {F} $}$_q$ (T); we establish a uniform lower bound for the canonical height of a point of \varphi, rational over the maximal abelian extension of K, and thus solve the so-called abelian version of the Lehmer problem in this situation. The classical original problem (a non torsion point in {$ \mathbb {G} $}$_m$ ({$ \mathbb {Q}$}$^{ab}$)) was solved by Amoroso and Dvornicich [1]. Soit \varphi un module de Drinfel'd d{\'e}fini sur une extension finie K de {$ \mathbb {F} $}$_q$ (T); nous d{\'e}montrons une minoration uniforme pour la hauteur canonique d'un point de \varphi, rationnel sur l'extension ab{\'e}lienne maximale de K, et r{\'e}solvons ainsi la version dite ab{\'e}lienne du probl{\`e}me de Lehmer dans cette situation. Dans le cadre classique (un point d'ordre infini de {$ \mathbb {G} $}$_m$ ({$ \mathbb {Q}$}$^{ab}$)), cette question a {\'e}t{\'e} r{\'e}solue par Amoroso et Dvornicich dans [1].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Anonymous:2008:AIV, author = "Anonymous", title = "Author Index (Volume 4)", journal = j-INT-J-NUMBER-THEORY, volume = "4", number = "6", pages = "1069--1072", month = dec, year = "2008", DOI = "https://doi.org/10.1142/S1793042108001900", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042108001900", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dewitt:2009:FGR, author = "Meghan Dewitt and Darrin Doud", title = "Finding {Galois} Representations Corresponding to Certain {Hecke} Eigenclasses", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "1--11", month = feb, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042109001888", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001888", abstract = "In 1992, Ash and McConnell presented computational evidence of a connection between three-dimensional Galois representations and certain arithmetic cohomology classes. For some examples, they were unable to determine the attached representation. For several Hecke eigenclasses (including one for which Ash and McConnell did not find the Galois representation), we find a Galois representation which appears to be attached and show strong evidence for the uniqueness of this representation. The techniques that we use to find defining polynomials for the Galois representations include a targeted Hunter search, class field theory and elliptic curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alaca:2009:NRP, author = "Ay{\c{s}}e Alaca and {\c{S}}aban Alaca and Mathieu F. Lemire and Kenneth S. Williams", title = "The Number of Representations of a Positive Integer by Certain Quaternary Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "13--40", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109001943", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001943", abstract = "Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x$^2$ + ey$^2$ + fz$^2$ + gt$^2$ with e, f, g \in {1, 2, 4, 8}.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Singh:2009:DPS, author = "Jitender Singh", title = "Defining power sums of $n$ and $ \phi (n)$ integers", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "41--53", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S179304210900189X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900189X", abstract = "Let n be a positive integer and \phi (n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those \phi (n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and \phi (n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Nathanson:2009:HFP, author = "Melvyn B. Nathanson", title = "Heights on the Finite Projective Line", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "55--65", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S179304210900192X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900192X", abstract = "Define the height function h(a) = {mink + (ka mod p) : k = 1, 2, \ldots, p - 1} for a \in {0, 1, \ldots, p - 1.} It is proved that the height has peaks at p, (p + 1)/2, and (p + c)/3, that these peaks occur at a = [p/3], (p - 3)/2, (p - 1)/2, [2p/3], p - 3, p 2, and p - 1, and that h(a) \leq p/3 for all other values of a.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Azaiez:2009:RHM, author = "Najib Ouled Azaiez", title = "Restrictions of {Hilbert} Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "67--80", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109001931", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001931", abstract = "Let \Gamma \subset PSL(2, {\mathbb{R}}) be a discrete and finite covolume subgroup. We suppose that the modular curve is ``embedded'' in a Hilbert modular surface, where \Gamma$_K$ is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (\rho$_n$)$_{n \in \mathbb {N} }$ of Hilbert modular forms for \Gamma$_K$ to modular forms for \Gamma. We denote by M$_{k 1}$, k$_2$ (\Gamma$_K$) the space of Hilbert modular forms of weight (k$_1$, k$_2$) for \Gamma$_K$. We prove that $ \sum_{n \in \mathbb {N} }$ $ \sum_{k 1}$, k$_2$ \in \mathbb{N} \rho$_n$ (M$_{k 1}$, k$_2$ (\Gamma$_K$)) is a subalgebra closed under Rankin--Cohen brackets of the algebra \oplus$_{k \in \mathbb {N} }$ M$_k$ (\Gamma) of modular forms for \Gamma.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Tanner:2009:SCP, author = "Noam Tanner", title = "Strings of Consecutive Primes in Function Fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "81--88", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109001918", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001918", abstract = "In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring {$ \mathbb {F} $}$_q$ [t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for {\em all\/} sufficiently large D.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Wiese:2009:MSC, author = "Gabor Wiese", title = "On Modular Symbols and the Cohomology of {Hecke} Triangle Surfaces", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "89--108", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109001967", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001967", abstract = "The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalized from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. A general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Weston:2009:PRF, author = "Tom Weston and Elena Zaurova", title = "Power Residues of {Fourier} Coefficients of Elliptic Curves with Complex Multiplication", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "109--124", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109001955", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001955", abstract = "Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{AlHajjShehadeh:2009:GFH, author = "Hala {Al Hajj Shehadeh} and Samar Jaafar and Kamal Khuri-Makdisi", title = "Generating Functions for {Hecke} Operators", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "125--140", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109001979", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001979", abstract = "Fix a prime N, and consider the action of the Hecke operator T$_N$ on the space of modular forms of full level and varying weight \kappa. The coefficients of the matrix of T$_N$ with respect to the basis {E$_4^i$ E$_6^j$ | 4i + 6j = \kappa } for can be combined for varying \kappa into a generating function F$_N$. We show that this generating function is a rational function for all N, and present a systematic method for computing F$_N$. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Rhoades:2009:SPD, author = "Robert C. Rhoades", title = "Statistics of Prime Divisors in Function Fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "141--152", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109001980", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001980", abstract = "We show that the prime divisors of a random polynomial in $ \mathbb {F}_q[t] $ are typically ``Poisson distributed''. This result is analogous to the result in {\mathbb{Z}} of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple proof of the Erd{\H{o}}s--Kac theorem in the function field setting. This approach gives stronger results about the moments of the sequence $ \omega (f)_{f \in { \mathbb {F} } q} [t] $ than was previously known, where $ \omega (f) $ is the number of prime divisors of $f$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Raji:2009:FCG, author = "Wissam Raji", title = "{Fourier} Coefficients of Generalized Modular Forms of Negative Weight", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "153--160", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002006", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002006", abstract = "The Fourier coefficients of classical modular forms of negative weights have been determined for the case for which F(\tau) belongs to a subgroup of the full modular group [9]. In this paper, we determine the Fourier coefficients of generalized modular forms of negative weights using the circle method.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Carr:2009:LIR, author = "Richard Carr and Cormac O'Sullivan", title = "On the Linear Independence of Roots", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "161--171", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002018", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002018", abstract = "A set of real nth roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kuo:2009:GST, author = "Wentang Kuo", title = "A Generalization of the {Sato--Tate Conjecture}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "1", pages = "173--184", month = feb, year = "2009", DOI = "https://doi.org/10.1142/S179304210900202X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:18 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900202X", abstract = "The original Sato--Tate Conjecture concerns the angle distribution of the eigenvalues arisen from non-CM elliptic curves. In this paper, we formulate an analogue of the Sato--Tate Conjecture on automorphic forms of (GL$_n$) and, under a holomorphic assumption, prove that the distribution is either uniform or the generalized Sato--Tate measure.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Rivoal:2009:AAI, author = "Tanguy Rivoal", title = "Applications arithm{\'e}tiques de l'interpolation lagrangienne. ({French}) [{Arithmetic} applications of {Lagrangianp} interpolation]", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "185--208", month = mar, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042109001992", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001992", abstract = "Newton's polynomial interpolation was applied in many situations in number theory, for example, to prove Polya's famous theorem on the growth of arithmetic entire function or the transcendency of e$^{\pi }$ by Gel'fond. In this paper, we study certain arithmetic applications of the rational interpolation defined by Ren{\'e} Lagrange in 1935, which was never done before. More precisely, we obtain new proofs of the irrationality of the numbers log(2) and \zeta (3). Furthermore, we provide a simultaneous generalization of Newton and Lagrange's interpolations, which enables us to get the irrationality of \zeta (2). L'interpolation polynomiale de Newton a eu de tr{\`e}s nombreuses applications arithm{\'e}tiques en th{\'e}orie des nombres, comme le c{\'e}l{\`e}bre th{\'e}or{\`e}me de Polya sur la croissance des fonctions enti{\`e}res arithm{\'e}tiques ou encore la transcendance de e$^{\pi }$ par Gel'fond. Dans ce papier, on pr{\'e}sente certaines applications arithm{\'e}tiques de l'interpolation rationnelle d{\'e}finie par Ren{\'e} Lagrange en 1935, ce qui n'avait jamais {\'e}t{\'e} fait auparavant. On retrouve ainsi l'irrationalit{\'e} des nombres log(2) et \zeta (3). On montre ensuite comment g{\'e}n{\'e}raliser simultan{\'e}ment l'interpolation de Newton et celle de Lagrange, ce qui nous permet de retrouver l'irrationalit{\'e} de \zeta (2).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Chaumont:2009:CSL, author = "Alain Chaumont and Johannes Leicht and Tom M{\"u}ller and Andreas Reinhart", title = "The Continuing Search for Large Elite Primes", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "209--218", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002031", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002031", abstract = "A prime number p is called {\em elite\/} if only finitely many Fermat numbers 2$^{2 n}$ + 1 are quadratic residues modulo p. So far, all 21 elite primes less than 250 billion were known, together with 24 larger items. We completed the systematic search up to the range of 2.5 \cdotp 10$^{12}$ finding six more such numbers. Moreover, 42 new elites larger than this bound were found, among which the largest has 374 596 decimal digits. A survey on the knowledge about elite primes together with some open problems and conjectures are presented.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Yee:2009:BPT, author = "Ae Ja Yee", title = "Bijective Proofs of a Theorem of {Fine} and Related Partition Identities", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "219--228", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002043", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002043", abstract = "In this paper, we prove a theorem of Fine bijectively. Stacks with summits and gradual stacks with summits are also discussed.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bandini:2009:CTE, author = "A. Bandini and I. Longhi", title = "Control Theorems for Elliptic Curves Over Function Fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "229--256", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002067", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002067", abstract = "Let F be a global field of characteristic p > 0, {$ \mathbb {F} $}/F a Galois extension with and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups Sel$_E$ (L)$_l$ (l any prime) as L varies through the subextensions of {$ \mathbb {F} $} via appropriate versions of Mazur's Control Theorem. In the case l = p, we let {$ \mathbb {F} $} = \cup {$ \mathbb {F} $}$_d$ where {$ \mathbb {F} $}$_d$ /F is a -extension. We prove that Sel$_E$ ({$ \mathbb {F} $}$_d$)$_p$ is a cofinitely generated {\mathbb{Z}}$_p$ [[Gal({\mathbb{Z}}$_d$ /F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic {$L$}-function associated to E in {\mathbb{Z}}$_p$ [[Gal({\mathbb{Z}}/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Murty:2009:SVP, author = "M. Ram Murty and N. Saradha", title = "Special Values of the Polygamma Functions", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "257--270", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002079", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002079", abstract = "Let q be a natural number and. We consider the Dirichlet series $ \sum_{n \geq 1} $ f(n)/n$^s$ and relate its value when s is a natural number, to the special values of the polygamma function. For certain types of functions f, we evaluate the special value explicitly and use this to study linear independence over {$ \mathbb {Q}$} of L(k,\chi) as \chi ranges over Dirichlet characters mod q which have the same parity as k.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chida:2009:IOS, author = "Masataka Chida", title = "Indivisibility of Orders of {Selmer} Groups for Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "271--280", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002080", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002080", abstract = "In this paper, we consider indivisibility of orders of Selmer groups for modular forms under quadratic twists. Then, we will give a generalization of a theorem of James--Ono and Kohnen--Ono.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kumchev:2009:BAE, author = "Angel V. Kumchev", title = "A Binary Additive Equation Involving Fractional Powers", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "281--292", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002092", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002092", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Li:2009:EPD, author = "Xian-Jin Li", title = "On the {Euler} Product of the {Dedekind} Zeta Function", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "293--301", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002109", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002109", abstract = "It is well known that the Euler product formula for the Riemann zeta function \zeta (s) is still valid for {\mathfrak{R}}(s) = 1 and s \neq 1. In this paper, we extend this result to zeta functions of number fields. In particular, we show that the Dedekind zeta function \zeta$_k$ (s) for any algebraic number field k can be written as the Euler product on the line {\mathfrak{R}}(s) = 1 except at the point s = 1. As a corollary, we obtain the Euler product formula on the line {\mathfrak{R}}(s) = 1 for Dirichlet {$L$}-functions L(s, \chi) of real characters.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Folsom:2009:CMU, author = "Amanda Folsom", title = "A Characterization of the Modular Units", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "303--310", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002110", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002110", abstract = "We provide an exact formula for the complex exponents in the modular product expansion of the modular units in terms of the Kubert--Lang structure theory, and deduce a characterization of the modular units in terms of the growth of these exponents, answering a question posed by Kohnen.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Nitaj:2009:CRCo, author = "Abderrahmane Nitaj", title = "Cryptanalysis of {RSA} with Constrained Keys", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "311--325", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002122", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002122", abstract = "Let n = pq be an RSA modulus with unknown prime factors of equal bit-size. Let e be the public exponent and d be the secret exponent satisfying ed \equiv 1 mod \varphi (n) where \varphi (n) is the Euler totient function. To reduce the decryption time or the signature generation time, one might be tempted to use a small private exponent d. Unfortunately, in 1990, Wiener showed that private exponents smaller than are insecure and in 1999, Boneh and Durfee improved the bound to n$^{0.292}$. In this paper, we show that instances of RSA with even large private exponents can be efficiently broken if there exist positive integers X, Y such that |eY - XF(u)| and Y are suitably small where F is a function of publicly known expression for which there exists an integer u \neq 0 satisfying F(u) \approx n and pu or qu is computable from F(u) and n. We show that the number of such exponents is at least O(n$^{3 / 4 - \varepsilon }$) when F(u) = p(q - u).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Andrews:2009:SIA, author = "George E. Andrews and Sylvie Corteel and Carla D. Savage", title = "On $q$-series identities arising from lecture hall partitions", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "327--337", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002134", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002134", abstract = "In this paper, we highlight two $q$-series identities arising from the ``five guidelines'' approach to enumerating lecture hall partitions and give direct, $q$-series proofs. This requires two new finite corollaries of a q-analog of Gauss's second theorem. In fact, the method reveals stronger results about lecture hall partitions and anti-lecture hall compositions that are only partially explained combinatorially.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pila:2009:EFS, author = "Jonathan Pila", title = "Entire Functions Sharing Arguments of Integrality, {I}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "339--353", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002146", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002146", abstract = "Let f be an entire function that is real and strictly increasing for all sufficiently large real arguments, and that satisfies certain additional conditions, and let X$_f$ be the set of non-negative real numbers at which f is integer valued. Suppose g is an entire function that takes integer values on X$_f$. We find growth conditions under which f,g must be algebraically dependent (over {\mathbb{Z}}) on X. The result generalizes a weak form of a theorem of P{\'o}lya.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Tanigawa:2009:FPM, author = "Yoshio Tanigawa and Wenguang Zhai", title = "On the fourth power moment of {$ \Delta x $} and {$ E(x) $} in short intervals", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "2", pages = "355--382", month = mar, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002055", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002055", abstract = "Let \Delta (x) and E(x) be error terms of the sum of divisor function and the mean square of the Riemann zeta function, respectively. In this paper, their fourth power moments for short intervals of Jutila's type are considered. We get an asymptotic formula for U in some range.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sands:2009:VFM, author = "Jonathan W. Sands", title = "Values at $ s = - 1 $ of {$L$}-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "383--405", month = may, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042109002183", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002183", abstract = "Fix a Galois extension of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in, let denote the primes of lying above those in S, and let denote the ring of -integers of. We then compare the Fitting ideal of as a {\mathbb{Z}}[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of {$ \mathbb {Q} $}[G], and hence in {\mathbb{Z}}[1/2][G]. Results in {\mathbb{Z}}[G] are obtained under the assumption of the Birch--Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where is a biquadratic extension of F containing the first layer of the cyclotomic {\mathbb{Z}}$_2$-extension of F, and describe a class of biquadratic extensions of F = {$ \mathbb {Q}$} that satisfy this condition.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baccar:2009:SSP, author = "N. Baccar and F. {Ben Sa{\"i}d}", title = "On Sets Such That the Partition Function Is Even from a Certain Point On", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "407--428", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002195", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002195", abstract = "Let P \in {$ \mathbb {F} $}$_2$ [z] with P(0) = 1 and degree(P) \geq 1. It is not difficult to prove (cf. [4,14]) that there is a unique subset of \mathbb{N} such that (mod 2), where denotes the number of partitions of n with parts in. However, finding the elements of such sets for general P seems to be hard. In this paper, we obtain solutions to this problem for a large class of polynomials P. Moreover, we give asymptotics for the counting function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chu:2009:ISH, author = "Wenchang Chu and Deyin Zheng", title = "Infinite Series with Harmonic Numbers and Central Binomial Coefficients", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "429--448", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002171", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib; http://www.math.utah.edu/pub/tex/bib/mathematica.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002171", abstract = "By means of two hypergeometric summation formulae, we establish two large classes of infinite series identities with harmonic numbers and central binomial coefficients. Up to now, these numerous formulae have hidden behind very few known identities of Ap{\'e}ry-like series for Riemann-zeta function, discovered mainly by Lehmer [14] and Elsner [12] as well as Borwein {\em et al.\/} [4, 5, 7]. All the computation and verification are carried out by an appropriately-devised {\em Mathematica\/} package.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ding:2009:SIF, author = "Shanshan Ding", title = "Smallest irreducible of the form $ x^2 - d y^2 $", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "449--456", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002158", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002158", abstract = "It is a classical result that prime numbers of the form x$^2$ + ny$^2$ can be characterized via class field theory for an infinite set of n. In this paper, we derive the function field analogue of the classical result. Then, we apply an effective version of the Chebotarev density theorem to bound the degree of the smallest irreducible of the form x$^2$- dy$^2$, where x, y, and d are elements of a polynomial ring over a finite field.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kominers:2009:CEE, author = "Scott Duke Kominers", title = "Configurations of Extremal Even Unimodular Lattices", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "457--464", month = may, year = "2009", DOI = "https://doi.org/10.1142/S179304210900216X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900216X", abstract = "We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Akbary:2009:RSM, author = "Amir Akbary and V. Kumar Murty", title = "Reduction $ \bmod p $ of subgroups of the {Mordell--Weil} group of an elliptic curve", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "465--487", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002225", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002225", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kurlberg:2009:PSS, author = "P{\"a}r Kurlberg", title = "{Poisson} Spacing Statistics for Value Sets of Polynomials", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "489--513", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002237", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002237", abstract = "If f is a non-constant polynomial with integer coefficients and q is an integer, we may regard f as a map from Z/qZ to Z/qZ. We show that the distribution of the (normalized) spacings between consecutive elements in the image of these maps becomes {\em Poissonian\/} as q tends to infinity along any sequence of square free integers such that the mean spacing modulo q tends to infinity.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Shavgulidze:2009:NRI, author = "Ketevan Shavgulidze", title = "On the Number of Representations of Integers by the Sums of Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "515--525", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002201", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002201", abstract = "We shall obtain the formulae for the number of representations of positive integers by a direct sum of k binary quadratic forms of the kind, when k = 3, 4, 5.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bosca:2009:PIA, author = "S{\'e}bastien Bosca", title = "Principalization of Ideals in {Abelian} Extensions of Number Fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "527--539", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002213", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002213", abstract = "We give a self-contained proof of a general conjecture of Gras on principalization of ideals in Abelian extensions of a given field L, which was solved by Kurihara in the case of totally real extensions L of the rational field {$ \mathbb {Q} $}. More precisely, for any given extension L/K of number fields, in which at least one infinite place of K totally splits, and for any ideal class c$_L$ of L, we construct a finite Abelian extension F/K, in which all infinite places totally split, such that c$_L$ become principal in the compositum M = LF.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sapar:2009:MEA, author = "S. H. Sapar and K. A. Mohd. Atan", title = "A method of estimating the $p$-adic sizes of common zeros of partial derivative polynomials associated with a quintic form", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "3", pages = "541--554", month = may, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002249", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:19 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002249", abstract = "It is known that the value of the exponential sum can be derived from the estimate of the cardinality |V|, the number of elements contained in the set where is the partial derivatives of with respect to. The cardinality of V in turn can be derived from the $p$-adic sizes of common zeros of the partial derivatives. This paper presents a method of determining the $p$-adic sizes of the components of (\xi, \eta) a common root of partial derivative polynomials of f(x,y) in $ Z_p$ [x,y] of degree five based on the $p$-adic Newton polyhedron technique associated with the polynomial. The degree five polynomial is of the form f(x,y) = ax$^5$ + bx$^4$ y + cx$^3$ y$^2$ + sx + ty + k. The estimate obtained is in terms of the $p$-adic sizes of the coefficients of the dominant terms in f.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kida:2009:QPH, author = "Masanari Kida and Gu{\'e}na{\"e}l Renault and Kazuhiro Yokoyama", title = "Quintic Polynomials of {Hashimoto--Tsunogai}, {Brumer} and {Kummer}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "555--571", month = jun, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042109002250", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002250", abstract = "We establish an isomorphism between the quintic cyclic polynomials discovered by Hashimoto--Tsunogai and those arising from Kummer theory for certain algebraic tori. This enables us to solve the isomorphism problem for Hashimoto--Tsunogai polynomials and also Brumer's quintic polynomials.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bringmann:2009:CDR, author = "Kathrin Bringmann", title = "Congruences for {Dyson}'s Ranks", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "573--584", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002262", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002262", abstract = "In this paper, we prove infinite families of congruences for coefficients of harmonic Maass forms whose coefficients encode Dyson's rank. This generalizes the earlier joint work of the author with Ken Ono.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alvanos:2009:CAC, author = "Paraskevas Alvanos and Yuri Bilu and Dimitrios Poulakis", title = "Characterizing Algebraic Curves with Infinitely Many Integral Points", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "585--590", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002274", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002274", abstract = "A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curveC over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper, we give necessary and sufficient conditions for C to have infinitely many S-integral points.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kozuma:2009:ECR, author = "Rintaro Kozuma", title = "Elliptic Curves Related to Cyclic Cubic Extensions", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "591--623", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002304", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002304", abstract = "The aim of this paper is to study certain family of elliptic curves defined over a number field F arising from hyperplane sections of some cubic surface associated to a cyclic cubic extension K/F. We show that each admits a 3-isogeny \varphi over F and the dual Selmer group is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve with nice property. We also prove that the Shafarevich--Tate group coincides with a class group of K as a special case.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Konyagin:2009:SPC, author = "Sergei V. Konyagin and Melvyn B. Nathanson", title = "Sums of Products of Congruence Classes and of Arithmetic Progressions", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "625--634", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002286", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002286", abstract = "Consider the congruence class R$_m$ (a) = {a + im : i \in Z} and the infinite arithmetic progression P$_m$ (a) = {a + im : i \in N$_0$ }. For positive integers a,b,c,d,m the sum of products set R$_m$ (a)R$_m$ (b) + R$_m$ (c)R$_m$ (d) consists of all integers of the form (a+im) \cdotp (b+jm)+(c+km)(d+\ell m) for some i,j,k,\ell \in Z. It is proved that if gcd(a,b,c,d,m) = 1, then R$_m$ (a)R$_m$ (b) + R$_m$ (c)R$_m$ (d) is equal to the congruence class R$_m$ (ab+cd), and that the sum of products set P$_m$ (a)P$_m$ (b)+P$_m$ (c)P$_m$ eventually coincides with the infinite arithmetic progression P$_m$ (ab+cd).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Oura:2009:EPA, author = "Manabu Oura", title = "{Eisenstein} Polynomials Associated to Binary Codes", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "635--640", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002298", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002298", abstract = "The Eisenstein polynomial is the weighted sum of the weight enumerators of all classes of Type II codes of fixed length. In this note, we investigate the ring generated by Eisenstein polynomials in genus 2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Moree:2009:PPA, author = "P. Moree and B. Sury", title = "Primes in a Prescribed Arithmetic Progression Dividing the Sequence", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "641--665", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002316", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002316", abstract = "Given positive integers a,b,c and d such that c and d are coprime, we show that the primes p \equiv c (mod d) dividing a$^k$ +b$^k$ for some k \geq 1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Tretkoff:2009:TSV, author = "Marvin D. Tretkoff and Paula Tretkoff", title = "Transcendence of Special Values of {Pochhammer} Functions", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "667--677", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S179304210900233X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900233X", abstract = "In this paper, we examine the set of algebraic numbers at which higher order hypergeometric functions take algebraic values. In particular, we deduce criteria for this set to be finite and for it to be infinite.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vulakh:2009:MSC, author = "L. Ya. Vulakh", title = "The {Markov} Spectra for Cocompact {Fuchsian} Groups", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "679--718", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002341", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002341", abstract = "Applying the Klein model D$^2$ of the hyperbolic plane and identifying the geodesics in D$^2$ with their poles in the projective plane, the author has developed a method for finding the discrete part of the Markov spectrum for Fuchsian groups. It is applicable mostly to non-cocompact groups. In the present paper, this method is extended to cocompact Fuchsian groups. For a group with signature (0;2,2,2,3), the complete description of the discrete part of the Markov spectrum is obtained. The result obtained leads to the complete description of the Markov and Lagrange spectra for the imaginary quadratic field with discriminant -20.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hofer:2009:DPG, author = "Roswitha Hofer and Peter Kritzer and Gerhard Larcher and Friedrich Pillichshammer", title = "Distribution properties of generalized {Van Der Corput--Halton} sequences and their subsequences", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "4", pages = "719--746", month = jun, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002328", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002328", abstract = "We study the distribution properties of sequences which are a generalization of the well-known van der Corput--Halton sequences on one hand, and digital (T,s)-sequences on the other. In this paper, we give precise results concerning the distribution properties of such sequences in the s-dimensional unit cube. Moreover, we consider subsequences of the above-mentioned sequences and study their distribution properties. Additionally, we give discrepancy estimates for some special cases, including subsequences of van der Corput and van der Corput--Halton sequences.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Adolphson:2009:ESI, author = "Alan Adolphson and Steven Sperber", title = "Exponential sums on {$ \mathbb {A}^n $}. {IV}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "747--764", month = aug, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042109002353", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002353", abstract = "We find new conditions on a polynomial over a finite field that guarantee that the exponential sum defined by the polynomial has only one nonvanishing $p$-adic cohomology group, hence the {$L$}-function associated to the exponential sum is a polynomial or the reciprocal of a polynomial.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cooper:2009:CES, author = "Shaun Cooper", title = "Construction of {Eisenstein} series for {$ \Gamma_0 (p) $}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "765--778", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002365", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002365", abstract = "A simple construction of Eisenstein series for the congruence subgroup \Gamma$_0$ (p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Salle:2009:MPG, author = "Landry Salle", title = "Mild pro-$p$-groups as {Galois} groups over global fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "779--795", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002377", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002377", abstract = "This paper is devoted to finding new examples of mild pro-p-groups as Galois groups over global fields, following the work of Labute ([6]). We focus on the Galois group of the maximal T-split S-ramified pro-p-extension of a global field k. We first retrieve some facts on presentations of such a group, including a study of the local-global principle for the cohomology group, then we show separately in the case of function fields and in the case of number fields how it can be used to find some mild pro-p-groups.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Byard:2009:TPQ, author = "Kevin Byard", title = "Tenth Power Qualified Residue Difference Sets", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "797--803", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002389", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002389", abstract = "Qualified residue difference sets of power n are known to exist for n = 2, 4, 6, as do similar sets that include the zero element, while both classes of set are known to be nonexistent for n = 8. Both classes of set are proved nonexistent for n = 10.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Taylor:2009:ACN, author = "Karen Taylor", title = "Analytic Continuation of Nonanalytic Vector-Valued {Eisenstein} Series", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "805--830", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002407", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002407", abstract = "In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin--Selberg convolution of a vector-valued modular cusp form associated to a monomial representation \rho of SL(2,{\mathbb{Z}}). This vector-valued Eisenstein series transforms under a representation \chi$_{\rho }$ associated to \rho. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jaafar:2009:M, author = "Samar Jaafar and Kamal Khuri-Makdisi", title = "On the maps from {$ X(4 p) $} to {$ X(4) $}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "831--844", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002390", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002390", abstract = "We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular forms give rise to a morphism F from the quotient of X(4p) by a certain involution \iota to projective space, such that F is a projective embedding of X(4p)/\iota away from the cusps. We also report on computer calculations regarding products of such modular forms, going up to weight 4 for p \leq 13, and up to weight 3 for p \leq 23, and make a conjecture about these products and the nature of the singularities at the cusps of the image F(X(4p)/\iota).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Knopp:2009:PGM, author = "Marvin Knopp and Geoffrey Mason", title = "Parabolic Generalized Modular Forms and Their Characters", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "845--857", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002419", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See revisions \cite{Knopp:2012:RPG}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002419", abstract = "We make a detailed study of the {\em generalized modular forms\/} of weight zero and their associated multiplier systems (characters) on an arbitrary subgroup \Gamma of finite index in the modular group. Among other things, we show that every generalized divisor on the compact Riemann surface associated to \Gamma is the divisor of a modular form (with {\em unitary\/} character) which is unique up to scalars. This extends a result of Petersson, and has applications to the Eichler cohomology.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Suarez:2009:MLC, author = "Ivan Suarez", title = "Modular Lattices Over {CM} Fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "859--869", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002420", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002420", abstract = "We study some properties of Arakelov-modular lattices, which are particular modular ideal lattices over CM fields. There are two main results in this paper. The first one is the determination of the number of Arakelov-modular lattices of fixed level over a given CM field provided that an Arakelov-modular lattice is already known. This number depends on the class numbers of the CM field and its maximal totally real subfield. The first part gives also a way to compute all these Arakelov-modular lattices. In the second part, we describe the levels that can occur for some multiquadratic CM number fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", keywords = "CM (complex multiplication)", } @Article{Angles:2009:WNC, author = "Bruno Angl{\`e}s and Tatiana Beliaeva", title = "On {Weil} Numbers in Cyclotomic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "871--884", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002432", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002432", abstract = "In this paper, we study the $p$-adic behavior of Weil numbers in the cyclotomic {\mathbb{Z}}$_p$-extension of the pth cyclotomic field. We determine the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in the Iwasawa theory. In a recent preprint [9] by Nguyen Quang Do and Nicolas, a generalization of this result to a semi-simple case was obtained.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dobi:2009:SRT, author = "Doris Dobi and Nicholas Wage and Irena Wang", title = "Supersingular Rank Two {Drinfel'd} Modules and Analogs of {Atkin}'s Orthogonal Polynomials", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "885--895", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002444", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002444", abstract = "The theory of elliptic curves and modular forms provides a precise relationship between the supersingular j-invariants and the congruences between modular forms. Kaneko and Zagier discuss a surprising generalization of these results in their paper on Atkin orthogonal polynomials. In this paper, we define an analog of the Atkin orthogonal polynomials for rank two Drinfel'd modules.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Glass:2009:RHC, author = "Darren Glass", title = "The $2$-Ranks of Hyperelliptic Curves with Extra Automorphisms", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "897--910", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002468", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002468", abstract = "This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring--Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p > 2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Delaunay:2009:SPE, author = "Christophe Delaunay and Christian Wuthrich", title = "Self-Points on Elliptic Curves of Prime Conductor", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "5", pages = "911--932", month = aug, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002456", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002456", abstract = "Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p], we construct a modular point P$_C$ on E, called self-point, as the image of (E,C) on X$_0$ (p) under the modular parametrization X$_0$ (p) \rightarrow E. We prove that the point is of infinite order in the Mordell--Weil group of E over the field of definition of C. One can deduce a lower bound on the growth of the rank of the Mordell--Weil group in its PGL$_2$ ({\mathbb{Z}}$_p$)-tower inside {$ \mathbb {Q}$}(E[p$^{\infty }$ ]).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Languasco:2009:CRE, author = "Alessandro Languasco", title = "A Conditional Result on the Exceptional Set for {Hardy--Littlewood} Numbers in Short Intervals", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "933--951", month = sep, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S179304210900247X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900247X", abstract = "Assuming the Generalized Riemann Hypothesis holds, we prove some conditional estimates on the exceptional set in short intervals for the Hardy--Littlewood problem.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bajnok:2009:MSS, author = "B{\'e}la Bajnok", title = "On the maximum size of a $ (k, l)$-sum-free subset of an abelian group", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "953--971", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002481", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002481", abstract = "A subset A of a given finite abelian group G is called (k,l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We are interested in finding the maximum size \lambda$_{k, l}$ (G) of a (k,l)-sum-free subset in G. A (2,1)-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group {\mathbb{Z}}$_n$ was found almost 40 years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of \lambda$_{3, 1}$ ({\mathbb{Z}}$_n$). More generally, a recent paper by Hamidoune and Plagne examines (k,l)-sum-free sets in G when $k - l$ and the order of G are relatively prime; we extend their results to see what happens without this assumption.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Luca:2009:MFP, author = "Florian Luca and Pantelimon St{\u{a}}nic{\u{a}}", title = "On {Machin}'s Formula with Powers of the {Golden} Section", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "973--979", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002493", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002493", abstract = "In this note, we find all solutions of the equation \pi /4 = a arctan(\varphi$^{\kappa }$) + b arctan(\varphi$^{\ell }$), in integers \kappa and \ell and rational numbers a and b, where \varphi is the golden section.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hegyvari:2009:ICL, author = "Norbert Hegyv{\'a}ri and Fran{\c{c}}ois Hennecart and Alain Plagne", title = "Iterated Compositions of Linear Operations on Sets of Positive Upper Density", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "981--997", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S179304210900250X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900250X", abstract = "Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded coefficients) on sets of integers having a positive upper density.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jones:2009:PVT, author = "Lenny Jones", title = "Polynomial Variations on a Theme of {Sierpi{\'n}ski}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "999--1015", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002511", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002511", abstract = "In 1960, Sierpi{\'n}ski proved that there exist infinitely many odd positive integers k such that k \cdotp 2$^n$ + 1 is composite for all integers n \geq 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) \in {\mathbb{Z}}[x], with f(1) \neq -d, such that f(x)x$^n$ + d is reducible over the rationals for all integers n \geq 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erd{\H{o}}s and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d \equiv 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d \equiv 6 (mod 12). We also consider the corresponding problem over {$ \mathbb {F} $}$_p$, and in that situation, we show 1 \in S for all primes p.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baier:2009:PQP, author = "Stephan Baier and Liangyi Zhao", title = "On Primes in Quadratic Progressions", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1017--1035", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002523", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002523", abstract = "We verify the Hardy--Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper by the authors [3].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dubickas:2009:FPR, author = "Art{\=u}ras Dubickas", title = "On the Fractional Parts of Rational Powers", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1037--1048", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002535", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002535", abstract = "Let \xi be a non-zero real number, and let a = p/q > 1 be a rational number. We denote by U(a,\xi) and L(a,\xi) the largest and the smallest limit points of the sequence of fractional parts {\xi a$^n$ }, n = 0,1,2,\ldots, respectively. A possible way to prove Mahler's conjecture claiming that Z-numbers do not exist is to show that U(3/2,\xi) > 1/2 for every \xi > 0. We prove that U(3/2,\xi) cannot belong to [0,1/3) \cup S, where S is an explicit infinite union of intervals in (1/3,1/2). This result is a corollary to a more general result claiming that, for any rational a > 1, U(a,\xi) cannot lie in a certain union of intervals. We also obtain new inequalities for the difference U(a,\xi) - L(a,\xi). Using them we show that some analogues of Z-numbers do not exist.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Knopp:2009:ECG, author = "Marvin Knopp and Joseph Lehner and Wissam Raji", title = "{Eichler} Cohomology for Generalized Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1049--1059", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002547", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002547", abstract = "By using Stokes's theorem, we prove an Eichler cohomology theorem for generalized modular forms with some restrictions on the relevant multiplier systems.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Katsurada:2009:DAA, author = "Masanori Katsurada and Takumi Noda", title = "Differential Actions on the Asymptotic Expansions of Non-Holomorphic {Eisenstein} Series", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1061--1088", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002559", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002559", abstract = "Let k be an arbitrary even integer, and E$_k$ (s;z) denote the non-holomorphic Eisenstein series (of weight k attached to SL$_2$ ({\mathbb{Z}})), defined by (1.1) below. In the present paper we first establish a complete asymptotic expansion of E$_k$ (s;z) in the descending order of y as y \rightarrow + \infty (Theorem 2.1), upon transferring from the previously derived asymptotic expansion of E$_0$ (s;z) (due to the first author [16]) to that of E$_k$ (s;z) through successive use of Maass' weight change operators. Theorem 2.1 yields various results on E$_k$ (s;z), including its functional properties (Corollaries 2.1--2.3), its relevant specific values (Corollaries 2.4--2.7), and its asymptotic aspects as z \rightarrow 0 (Corollary 2.8). We then apply the non-Euclidean Laplacian \Delta$_{H, k}$ (of weight k attached to the upper-half plane) to the resulting expansion, in order to justify the eigenfunction equation for E$_k$ (s;z) in (1.6), where the justification can be made uniformly in the whole s-plane (Theorem 2.2).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pong:2009:LSN, author = "Wai Yan Pong", title = "Length Spectra of Natural Numbers", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1089--1102", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002584", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002584", abstract = "A natural number n can generally be written as a sum of m consecutive natural numbers for various values of m \geq 1. The length spectrum of n is the set of these admissible m. Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute the equivalence classes of this relation. Moreover, we show that these classes can only have either 1,2 or infinitely many elements.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pries:2009:TCL, author = "Rachel Pries", title = "The $p$-torsion of curves with large $p$-rank", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1103--1116", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002560", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002560", abstract = "Consider the moduli space of smooth curves of genus g and $p$-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the $p$-torsion of the Jacobians of these curves for f < g - 1. We prove that the generic point of every component of this moduli space has a-number 1 when f = g - 2 and f = g - 3. Likewise, we show that a generic hyperelliptic curve with $p$-rank g 2 has a-number 1 when p \geq 3. We also show that the locus of curves with $p$-rank g - 2 and a-number 2 is non-empty with codimension 3 in when p \geq 5. We include some other results when f = g - 3. The proofs are by induction on g while fixing g - f. They use computations about certain components of the boundary of.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Muriefah:2009:DE, author = "Fadwa S. Abu Muriefah and Florian Luca and Samir Siksek and Szabolcs Tengely", title = "On the {Diophantine} equation {$ x^2 + C = 2 y^n $}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1117--1128", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002572", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002572", abstract = "In this paper, we study the Diophantine equation x$^2$ + C = 2y$^n$ in positive integers x,y with gcd(x,y) = 1, where n \geq 3 and C is a positive integer. If C \equiv 1 (mod 4), we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequences due to Bilu, Hanrot and Voutier. We illustrate our approach by solving completely the equations x$^2$ + 17$^{a 1}$ = 2y$^n$, x$^2$ + 5$^{a 1}$ 13$^{a 2}$ = 2y$^n$ and x$^2$ + 3$^{a 1}$ 11$^{a 2}$ = 2y$^n$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chen:2009:GSP, author = "Sin-Da Chen and Sen-Shan Huang", title = "On General Series--Product Identities", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "6", pages = "1129--1148", month = sep, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002596", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:20 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002596", abstract = "We derive the general series--product identities from which we deduce several applications, including an identity of Gauss, the generalization of Winquist's identity by Carlitz and Subbarao, an identity for, the quintuple product identity, and the octuple product identity.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Liu:2009:GRT, author = "Yu-Ru Liu and Craig V. Spencer", title = "A Generalization of {Roth}'s Theorem in Function Fields", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1149--1154", month = nov, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042109002602", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002602", abstract = "Let {$ \mathbb {F} $}$_q$ [t] denote the polynomial ring over the finite field {$ \mathbb {F} $}$_q$, and let denote the subset of {$ \mathbb {F} $}$_q$ [t] containing all polynomials of degree strictly less than N. For non-zero elements r$_1$, \ldots, r$_s$ of {$ \mathbb {F} $}$_q$ satisfying r$_1$ + \cdots + r$_s$ = 0, let denote the maximal cardinality of a set which contains no non-trivial solution of r$_1$ x$_1$ + \cdots + r$_s$ x$_s$ = 0 with x$_i$ \in A (1 \leq i \leq s). We prove that.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bayer-Fluckiger:2009:EMC, author = "Eva Bayer-Fluckiger and Jean-Paul Cerri and J{\'e}r{\^o}me Chaubert", title = "{Euclidean} Minima and Central Division Algebras", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1155--1168", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002614", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002614", abstract = "The notion of Euclidean minimum of a number field is a classical one. In this paper, we generalize it to central division algebras and establish some general results in this new context.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Liu:2009:NLT, author = "Huaning Liu", title = "A note on {Lehmer} $k$-tuples", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1169--1178", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002626", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002626", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kuo:2009:GLD, author = "Wentang Kuo and Yu-Ru Liu", title = "{Gaussian} Laws on {Drinfeld} Modules", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1179--1203", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002638", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002638", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Belliard:2009:ACC, author = "Jean-Robert Belliard", title = "Asymptotic Cohomology of Circular Units", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1205--1219", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S179304210900264X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304210900264X", abstract = "Let F be a number field, abelian over {$ \mathbb {Q} $}, and fix a prime p \neq 2. Consider the cyclotomic {\mathbb{Z}}$_p$-extension F$_{\infty }$ /F and denote F$_n$ the nth finite subfield and C$_n$ its group of circular units as defined by Sinnott. Then the Galois groups G$_{m, n}$ = Gal(F$_m$ /F$_n$) act naturally on the C$_m$ 's (for any m \geq n \gg 0). We compute the Tate cohomology groups for i = -1,0 without assuming anything else neither on F nor on p.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Olofsson:2009:LRH, author = "Rikard Olofsson", title = "Local {Riemann} Hypothesis for Complex Numbers", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1221--1230", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002651", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002651", abstract = "In this paper, a special class of local \zeta functions is studied. The main theorem states that the functions have all zeros on the line {\mathfrak{R}}(s) = 1/2. This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have {\mathfrak{R}}(s) = 1/2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bundschuh:2009:ARC, author = "Peter Bundschuh and Keijo V{\"a}{\"a}n{\"a}nen", title = "Arithmetical results on certain $q$-series, {II}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1231--1245", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002663", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002663", abstract = "As in Part I, entire transcendental solutions of certain mth order linear q-difference equations are investigated arithmetically, where now the polynomial coefficients are much more general. The purpose of this paper is to produce again lower bounds for the dimension of the K-vector space generated by 1 and the values of these solutions at m successive powers of q, where K is the rational or an imaginary quadratic field. A new feature in the proof is to use simultaneously positive and negative powers of q as interpolation points leading to an extra parameter in the main result extending its applicability.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2009:COD, author = "Ping-Shun Chan and Yuval Z. Flicker", title = "Cyclic Odd Degree Base Change Lifting for Unitary Groups in Three Variables", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1247--1309", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002687", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002687", abstract = "Let F be a number field or a $p$-adic field of odd residual characteristic. Let E be a quadratic extension of F, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (respectively, admissible) representations from the unitary group U(3, E/F) to the unitary group U(3, F' E/F'). As a consequence, we classify, up to certain restrictions, the packets of U(3, F' E/F') which contain irreducible automorphic (respectively, admissible) representations invariant under the action of the Galois group Gal(F' E/E). We also determine the invariance of individual representations. This work is the first study of base change into an algebraic group whose packets are not all singletons, and which does not satisfy the rigidity, or ``strong multiplicity one'', theorem. Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (respectively, admissible) member is Galois-invariant. The restriction that the residual characteristic of the local fields be odd may be removed once the multiplicity one theorem for U(3) is proved to hold unconditionally without restriction on the dyadic places.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alladi:2009:NCS, author = "Krishnaswami Alladi", title = "A new combinatorial study of the {Rogers--Fine} identity and a related partial theta series", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1311--1320", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002675", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002675", abstract = "We provide a transparent combinatorial derivation of a variant of the Rogers--Fine identity and a new combinatorial proof of a related partial theta series.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dummigan:2009:SSF, author = "Neil Dummigan", title = "Symmetric Square {$L$}-Functions and {Shafarevich--Tate} Groups, {II}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "7", pages = "1321--1345", month = nov, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002699", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002699", abstract = "We re-examine some critical values of symmetric square {$L$}-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich--Tate groups, demanded by the Bloch--Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen--Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer--Ibukiyama generalization of the Rankin--Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch--Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard {$L$}-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Toulmonde:2009:CAV, author = "Vincent Toulmonde", title = "Comportement au voisinage de $1$ de la fonction de r{\'e}partition de $ \phi (n) / n$. ({French}) [{Behavior} in the neighborhood of $1$ of the partition function $ \phi (n) / n $]", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1347--1384", month = dec, year = "2009", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042109001414", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109001414", abstract = "Let \phi denote Euler's totient function, and G be the distribution function of \phi (n)/n. Using functional equations, it is shown that \phi (n)/n is statistically close to 1 essentially when prime factors of n are large. A function defined by a difference-differential equation gives a quantitative measure of the statistical influence of the size of prime factors of n on the closeness of \phi (n)/n to 1. As a corollary, an asymptotic expansion at any order of G(1)-G(1-\varepsilon) is obtained according to negative powers of log(1/\varepsilon), when \varepsilon tends to 0$^+$. This improves a result of Erd{\H{o}}s (1946) in which he gives the first term of it. By optimally choosing the order of this expansion, an estimation of G(1)-G(1-\varepsilon) is deduced, involving an error term of the same size as the best known error term involved in prime number theorem. Soit \phi l'indicatrice d'Euler. Nous {\'e}tudions le comportement au voisinage de 1 de la fonction G de r{\'e}partition de \phi (n)/n, via la mise en {\'e}vidence d'{\'e}quations fonctionnelles. Nous obtenons un r{\'e}sultat mesurant l'influence statistique de la taille du plus petit facteur premier d'un entier g{\'e}n{\'e}rique n quant {\`a} la proximit{\'e} de \phi (n)/n par rapport {\`a} 1. Ce r{\'e}sultat met en jeu une fonction d{\'e}finie par une {\'e}quation diff{\'e}rentielle aux diff{\'e}rences. Nous en d{\'e}duisons un d{\'e}veloppement limit{\'e} {\`a} tout ordre de G(1)-G(1-\varepsilon ) selon les puissances de 1/(log 1/\varepsilon), am{\'e}liorant ainsi un r{\'e}sultat d'Erd{\H{o}}s (1946) dans lequel il obtient le premier terme de ce d{\'e}veloppement. Une troncature convenable de ce d{\'e}veloppement fournit un terme d'erreur comparable {\`a} celui actuellement connu pour le th{\'e}or{\`e}me des nombres premiers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Berkovich:2009:TPN, author = "Alexander Berkovich", title = "The Tri-Pentagonal Number Theorem and Related Identities", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1385--1399", month = dec, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002705", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002705", abstract = "I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater list. Finally, I establish a new infinite family of multiple series identities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ressler:2009:BQF, author = "Wendell Ressler", title = "On Binary Quadratic Forms and the {Hecke} Groups", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1401--1418", month = dec, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002730", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002730", abstract = "We present a reduction theory for certain binary quadratic forms with coefficients in {\mathbb{Z}}[\lambda ], where \lambda is the minimal translation in a Hecke group. We generalize from the modular group \Gamma (1) = PSL(2,{\mathbb{Z}}) to the Hecke groups and make extensive use of modified negative continued fractions. We also define and characterize ``reduced'' and ``simple'' hyperbolic fixed points of the Hecke groups.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bell:2009:BSF, author = "Jason P. Bell and Jonathan W. Bober", title = "Bounded Step Functions and Factorial Ratio Sequences", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1419--1431", month = dec, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002742", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002742", abstract = "We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a restriction on when such a factorial ratio sequence can be integral. Additionally, we note that this work has applications to the classification of cyclic quotient singularities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{El-Guindy:2009:FEM, author = "Ahmad El-Guindy", title = "{Fourier} Expansions with Modular Form Coefficients", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1433--1446", month = dec, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002717", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002717", abstract = "In this paper, we study the Fourier expansion where the coefficients are given as the evaluation of a sequence of modular forms at a fixed point in the upper half-plane. We show that for prime levels l for which the modular curve X$_0$ (l) is hyperelliptic (with hyperelliptic involution of the Atkin--Lehner type) then one can choose a sequence of weight k (any even integer) forms so that the resulting Fourier expansion is itself a meromorphic modular form of weight 2-k. These sequences have many interesting properties, for instance, the sequence of their first nonzero next-to-leading coefficient is equal to the terms in the Fourier expansion of a certain weight 2-k form. The results in the paper generalizes earlier work by Asai, Kaneko, and Ninomiya (for level one), and Ahlgren (for the cases where X$_0$ (l) has genus zero).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ehrenpreis:2009:EPS, author = "Leon Ehrenpreis", title = "{Eisenstein} and {Poincar{\'e}} Series on {$ \mathrm {SL}(3, r) $}", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1447--1475", month = dec, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002729", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002729", abstract = "This work continues the ideas presented in the author's book, {\em The Universality of the Radon Transform\/} (Oxford, 2003), which deals with the group SL(2,R). The complication that arises for G = SL(3,R) comes from the fact that there are now two fundamental representations. This has the consequence that the wave operator, which plays a central role in our work on SL(2,R) is replaced by an overdetermined system of partial differential equations. The analog of the wave operator is defined using an MN invariant orbit of G acting on the direct sum of the symmetric squares of the fundamental representations. The relation of orbits, or, in general, of any algebraic variety, to a system of partial differential equations comes via the Fundamental Principle, which shows how Fourier transforms of functions or measures on an algebraic variety correspond to solutions of the system of partial differential equations defined by the equations of the variety. In particular, we can start with the sum T of the delta functions of the orbit of the group \Gamma = SL(2,Z) on the light cone. We then take its Fourier transform, using a suitable quadratic form. We then decompose the Fourier transform under the commuting group of G. In this way, we obtain a \Gamma invariant distribution which has a natural restriction to the orbit G/K, which is the symmetric space of G. This restriction is (essentially) the nonanalytic Eisenstein series. We can compute the periods of the Eisenstein series over various orbits of subgroups of G by means of the Euclidean Plancherel formula. A more complicated form of these ideas is needed to define Poincar{\'e} series.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Liu:2009:SFT, author = "Zhi-Guo Liu and Xiao-Mei Yang", title = "On the {Schr{\"o}ter} formula for theta functions", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1477--1488", month = dec, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002754", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002754", abstract = "The Schr{\"o}ter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schr{\"o}ter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Anonymous:2009:AIV, author = "Anonymous", title = "Author Index (Volume 5)", journal = j-INT-J-NUMBER-THEORY, volume = "5", number = "8", pages = "1489--1493", month = dec, year = "2009", DOI = "https://doi.org/10.1142/S1793042109002766", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042109002766", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baoulina:2010:NSC, author = "Ioulia Baoulina", title = "On the Number of Solutions to Certain Diagonal Equations Over Finite Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "1--14", month = feb, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110002776", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002776", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Munshi:2010:DPR, author = "Ritabrata Munshi", title = "Density of Positive Rank Fibers in Elliptic Fibrations, {II}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "15--23", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002867", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002867", abstract = "We show that for a quartic elliptic fibration over a real number field, existence of two positive rank fibers implies existence of a dense set of positive rank fibers. We also prove the same result for certain sextic families.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Krieg:2010:TSH, author = "Aloys Krieg", title = "Theta Series Over the {Hurwitz} Quaternions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "25--36", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002788", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002788", abstract = "There are six theta constants over the Hurwitz quaternions on the quaternion half-space of degree 2. The paper describes the behavior of these theta constants under the transpose mapping, which can be derived from the Fourier expansions. The results are applied to the theta series of the first and second kind.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alaca:2010:FOQ, author = "Ay{\c{s}}e Alaca and {\c{S}}aban Alaca and Kenneth S. Williams", title = "Fourteen Octonary Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "37--50", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S179304211000279X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000279X", abstract = "We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Beck:2010:FTC, author = "Matthias Beck and Mary Halloran", title = "Finite Trigonometric Character Sums Via Discrete {Fourier} Analysis", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "51--67", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002806", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002806", abstract = "We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were first systematically studied by Berndt and Zaharescu where their proofs involved complex contour integration. We show how to prove most of Berndt--Zaharescu's and some new identities by elementary methods of discrete Fourier analysis.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Miller:2010:ATN, author = "Alison Miller and Aaron Pixton", title = "Arithmetic Traces of Non-Holomorphic Modular Invariants", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "69--87", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002818", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002818", abstract = "We extend results of Bringmann and Ono that relate certain generalized traces of Maass--Poincar{\'e} series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences and. We show that if f is a modular form of non-positive weight 2 - 2 \lambda and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2010:CSA, author = "Heng Huat Chan and Shaun Cooper and Francesco Sica", title = "Congruences Satisfied by {Ap{\'e}ry}-Like Numbers", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "89--97", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002879", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002879", abstract = "In this article, we investigate congruences satisfied by Ap{\'e}ry-like numbers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hassen:2010:HZF, author = "Abdul Hassen and Hieu D. Nguyen", title = "Hypergeometric Zeta Functions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "99--126", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S179304211000282X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000282X", abstract = "This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kane:2010:RIT, author = "Ben Kane", title = "Representations of Integers by Ternary Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "127--158", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002831", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002831", abstract = "We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen's plus space, where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for {$L$}-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan--Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", }

%% Check page gap at publisher site: v6 n1 pp159--160

@Article{Bremner:2010:CST, author = "Andrew Bremner and Blair K. Spearman", title = "Cyclic sextic trinomials {$ x^6 + A x + B $}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "161--167", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002843", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002843", abstract = "A correspondence is obtained between irreducible cyclic sextic trinomials x$^6$ + Ax + B \in {$ \mathbb {Q}$}[x] and rational points on a genus two curve. This implies that up to scaling, x$^6$ + 133x + 209 is the only cyclic sextic trinomial of the given type.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Alaca:2010:SQF, author = "Ay{\c{s}}e Alaca and {\c{S}}aban Alaca and Kenneth S. Williams", title = "Sextenary Quadratic Forms and an Identity of {Klein} and {Fricke}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "169--183", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002880", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002880", abstract = "Formulae, originally conjectured by Liouville, are proved for the number of representations of a positive integer n by each of the eight sextenary quadratic forms, , , , , , , .", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Boylan:2010:APC, author = "Matthew Boylan", title = "Arithmetic Properties of Certain Level One {Mock} Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "185--202", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002855", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002855", abstract = "In a recent work, Bringmann and Ono [4] show that Ramanujan's f(q) mock theta function is the holomorphic projection of a harmonic weak Maass form of weight 1/2. In this paper, we extend the work of Ono in [13]. In particular, we study holomorphic projections of certain integer weight harmonic weak Maass forms on SL$_2$ ({\mathbb{Z}}) using Hecke operators and the differential theta-operator.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Copil:2010:RCP, author = "Vlad Copil and Lauren{\c{t}}iu Panaitopol", title = "On the Ratio of Consecutive Primes", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "203--210", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002934", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002934", abstract = "For n \geq 1, let p$_n$ be the nth prime number and for n \geq 1. Using several results of Erd{\H{o}}s, we study the sequence (q$_n$)$_{n \geq 1}$ and we prove similar results as for the sequence (d$_n$)$_{n \geq 1}$, d$_n$ = p$_{n + 1}$- p$_n$. We also consider the sequence for n \geq 1 and denote by G$_n$ and A$_n$ its geometrical and arithmetical averages. We prove that for n \geq 4.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Coons:2010:TSR, author = "Michael Coons", title = "The Transcendence of Series Related to {Stern}'s Diatomic Sequence", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "1", pages = "211--217", month = feb, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002958", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:21 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002958", abstract = "We prove various transcendence results regarding the Stern sequence and related functions; in particular, we prove that the generating function of the Stern sequence is transcendental. Transcendence results are also proven for the generating function of the Stern polynomials and for power series whose coefficients arise from some special subsequences of Stern's sequence.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lagarias:2010:CSS, author = "Jeffrey C. Lagarias", title = "Cyclic Systems of Simultaneous Congruences", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "219--245", month = mar, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110002892", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See erratum \cite{Lagarias:2010:ECS}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002892", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kim:2010:BPQ, author = "Sun Kim", title = "A Bijective Proof of the Quintuple Product Identity", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "247--256", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002909", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002909", abstract = "We give a bijective proof of the quintuple product identity using bijective proofs of Jacobi's triple product identity and Euler's recurrence relation.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bouallegue:2010:KNS, author = "Kais Bouall{\`e}gue and Othman Echi and Richard G. E. Pinch", title = "{Korselt} Numbers and Sets", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "257--269", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002922", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002922", abstract = "Let $ \alpha \in \mathbb {Z} \setminus {0} $. A positive integer $N$ is said to be an $ \alpha $-Korselt number ($ K_{\alpha }$-number, for short) if $ N \neq \alpha $ and $ p \alpha $ divides $ N - \alpha $ for each prime divisor $p$ of $N$. We are concerned, here, with both a numerical and theoretical study of composite squarefree Korselt numbers. The paper contains two main results. The first one shows that for $ \alpha \in \mathbb {Z} \setminus {0}$, the following properties hold: (i) If $ \alpha \leq 1$, then each composite squarefree $ K_{\alpha }$-number has at least three prime factors. (ii) Suppose that $ \alpha > 1$. Let $ p < q$ be two prime numbers and $ N \coloneq p q$. If $N$ is an \alpha Korselt number, then $ p < q \leq 4 \alpha - 3$. In particular, there are only finitely many $ \alpha $ Korselt numbers with exactly two prime factors. Let $ \alpha \in \mathbb {N} \setminus {0}$; by an $ \alpha $-Williams number ($ W_{\alpha }$-number, for short) we mean a positive integer which is both a $ K_{\alpha }$-number and a $ K_{- \alpha }$-number. Our second main result shows that if $p$, $ 3 p - 2 $, $ 3 p + 2$ are all prime, then their product is a ($ 3 p$)-Williams number.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kelmer:2010:DTK, author = "Dubi Kelmer", title = "Distribution of Twisted {Kloosterman} Sums Modulo Prime Powers", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "271--280", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002910", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002910", abstract = "In this paper, we study Kloosterman sums twisted by multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Garvan:2010:CAS, author = "F. G. Garvan", title = "Congruences for {Andrews}' Smallest Parts Partition Function and New Congruences for {Dyson}'s Rank", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "281--309", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S179304211000296X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000296X", abstract = "Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt(n) mod \ell for \ell = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bennett:2010:DE, author = "Michael A. Bennett and Jordan S. Ellenberg and Nathan C. Ng", title = "The {Diophantine} equation {$ A^4 + 2^\delta B^2 = C^n $}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "311--338", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002971", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002971", abstract = "In a previous paper, the second author proved that the equation $ A^4 + B^2 = C^p $ had no integral solutions for prime $ p > 211 $ and $ (A, B, C) = 1 $. In the present paper, we explain how to extend this result to smaller exponents, and to the related equation $ A^4 + 2 B^2 = C^p $.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Souderes:2010:MDS, author = "Ismael Soud{\`e}res", title = "{Motivic} Double Shuffle", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "339--370", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002995", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002995", abstract = "The goal of this paper is to give an elementary proof of the double shuffle relations directly for the Goncharov and Manin motivic multiple zeta values. The shuffle relation is straightforward, but for the stuffle, we use a modification of a method first introduced by Cartier for the purpose of proving stuffle for the real multiple zeta values. We will use both the representation of multiple zeta values on the moduli spaces of curve introduced by Goncharov and Manin and we will apply suitable blow-up sequences to the representation of multiple zeta values as integral over a cube.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Han:2010:FNP, author = "Jeong Soon Han and Hee Sik Kim and J. Neggers", title = "The {Fibonacci}-Norm of a Positive Integer: Observations and Conjectures", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "371--385", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003009", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See acknowledgement of priority \cite{Han:2011:APF}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003009", abstract = "In this paper, we define the Fibonacci-norm of a natural number n to be the smallest integer k such that n|F$_k$, the kth Fibonacci number. We show that for m \geq 5. Thus by analogy we say that a natural number n \geq 5 is a local-Fibonacci-number whenever . We offer several conjectures concerning the distribution of local-Fibonacci-numbers. We show that, where provided F$_{m + k}$ \equiv F$_m$ (mod n) for all natural numbers m, with k \geq 1 the smallest natural number for which this is true.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Nebe:2010:LDS, author = "Gabriele Nebe and Boris Venkov", title = "Low-Dimensional Strongly Perfect Lattices. {III}: Dual Strongly Perfect Lattices of Dimension $ 14 $", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "387--409", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003022", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003022", abstract = "A lattice is called dual strongly perfect if both, the lattice and its dual, are strongly perfect. We show that the extremal 3-modular lattice [\pm G$_2$ (3)]$_{14}$ with automorphism group C$_2$ $ \times $ G$_2$ ({$ \mathbb {F} $}$_3$) is the unique dual strongly perfect lattice of dimension 14.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vulakh:2010:MBI, author = "L. Ya. Vulakh", title = "Minima of Binary Indefinite {Hermitian} Forms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "411--435", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003034", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003034", abstract = "Classification of binary indefinite primitive Hermitian forms modulo the action of the extended Bianchi group (or Hilbert modular group) B$_d$ is given. When the discriminant of the quadratic field (and d) is negative, the results obtained can be applied to classify the maximal non-elementary Fuchsian subgroups of B$_d$, and to find the Hermitian points in the Markov spectrum of B$_d$. If \nu is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H$^3$, the upper half-space model of the three-dimensional hyperbolic space, with diameter 1/\nu, which depends on one continuous parameter.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zhao:2010:EDP, author = "Yusheng Zhao and Wei Li and Xianke Zhang", title = "Effective Determination of Prime Decompositions of Cubic Function Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "437--448", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110002983", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002983", abstract = "In this paper, we determine completely the prime decomposition of cubic function fields by effective and explicit methods.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kim:2010:CPC, author = "Byungchan Kim", title = "Combinatorial Proofs of Certain Identities Involving Partial Theta Functions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "2", pages = "449--460", month = mar, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003046", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003046", abstract = "In this brief note, we give combinatorial proofs of two identities involving partial theta functions. As an application, we prove an identity for the product of partial theta functions, first established by Andrews and Warnaar. We also provide a generalization of the first two identities and give a combinatorial proof of the generalized identities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{McCarthy:2010:HSP, author = "Dermot McCarthy", title = "{$_3 F_2$} Hypergeometric Series and Periods of Elliptic Curves", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "461--470", month = may, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110002946", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110002946", abstract = "We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Viada:2010:LBN, author = "Evelina Viada", title = "Lower Bounds for the Normalized Height and Non-Dense Subsets of Subvarieties of {Abelian} Varieties", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "471--499", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003010", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003010", abstract = "This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup \Gamma of finite rank, are non-Zariski-dense in V. If \Gamma has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Borwein:2010:DTM, author = "Jonathan M. Borwein and O-Yeat Chan", title = "Duality in tails of multiple-zeta values", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "501--514", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003058", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", MRclass = "11M32 (33C20 33F05)", MRnumber = "2652893", MRreviewer = "Zhonghua Li", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "http://docserver.carma.newcastle.edu.au/1218/; https://www.worldscientific.com/doi/10.1142/S1793042110003058", abstract = "Duality relations are deduced for tails of multiple-zeta values using elementary methods. These formulas extend the classical duality formulas for multiple-zeta values.", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", researcherid-numbers = "Borwein, Jonathan/A-6082-2009", unique-id = "Borwein:2010:DTM", } @Article{Chu:2010:BBL, author = "Wenchang Chu and Wenlong Zhang", title = "Bilateral {Bailey} Lemma and False Theta Functions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "515--577", month = may, year = "2010", DOI = "https://doi.org/10.1142/S179304211000306X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000306X", abstract = "By examining the transformation formulae between unilateral series and bilateral ones derived from the bilateral Bailey lemma, we establish numerous identities of false theta functions, including most of the known ones discovered mainly by Rogers [40] and Ramanujan in his \booktitle{Lost Notebook} [39].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Fehm:2010:RAV, author = "Arno Fehm and Sebastian Petersen", title = "On the Rank of {Abelian} Varieties Over Ample Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "579--586", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003071", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003071", abstract = "A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the {\mathbb{Z}}$_{(p)}$-module A(K) \otimes {\mathbb{Z}}$_{(p)}$ is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bugeaud:2010:PRS, author = "Yann Bugeaud and Maurice Mignotte", title = "Polynomial Root Separation", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "587--602", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003083", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003083", abstract = "We discuss the following question: How close to each other can two distinct roots of an integer polynomial be? We summarize what is presently known on this and related problems, and establish several new results on root separation of monic, integer polynomials.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ruhl:2010:AIQ, author = "Klaas-Tido R{\"u}hl", title = "Annihilating Ideals of Quadratic Forms Over Local and Global Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "603--624", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003095", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003095", abstract = "We study annihilating polynomials and annihilating ideals for elements of Witt rings for groups of exponent 2. With the help of these results and certain calculations involving the Clifford invariant, we are able to give full sets of generators for the annihilating ideal of both the isometry class and the equivalence class of an arbitrary quadratic form over a local field. By applying the Hasse--Minkowski theorem, we can then achieve the same for an arbitrary quadratic form over a global field.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Le:2010:APT, author = "Daniel Le and Shelly Manber and Shrenik Shah", title = "On $p$-adic properties of twisted traces of singular moduli", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "625--653", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003101", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003101", abstract = "We prove that logarithmic derivatives of certain twisted Hilbert class polynomials are holomorphic modular forms modulo p of filtration p + 1. We derive $p$-adic information about twisted Hecke traces and Hilbert class polynomials. In this framework, we formulate a precise criterion for $p$-divisibility of class numbers of imaginary quadratic fields in terms of the existence of certain cusp forms modulo p. We explain the existence of infinite classes of congruent twisted Hecke traces with fixed discriminant in terms of the factorization of the associated Hilbert class polynomial modulo p. Finally, we provide a new proof of a theorem of Ogg classifying those p for which all supersingular j-invariants modulo p lie in F$_p$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Haynes:2010:NDN, author = "Alan K. Haynes", title = "Numerators of Differences of Nonconsecutive {Farey} Fractions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "655--666", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003113", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003113", abstract = "An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions, the numerators of the differences are well understood and have applications to several interesting problems. In this paper, we investigate numerators of differences of fractions which are farther apart. We establish algebraic identities between such differences which then allow us to calculate their average values by using properties of a measure preserving transformation of the Farey triangle.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Witno:2010:EPP, author = "Amin Witno", title = "On Elite Primes of Period Four", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "667--671", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003149", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003149", abstract = "A prime p is elite if all sufficiently large Fermat numbers F$_n$ = 2$^{2 n}$ + 1 are quadratic nonresidues modulo p. In contrast, p is anti-elite if all sufficiently large F$_n$ are quadratic residues modulo p. The sequence F$_n$ modulo p is necessarily periodic. We give a sequence of pairwise coprime integers whose prime factors are each elite or anti-elite with period four.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2010:RCCa, author = "Hei-Chi Chan", title = "{Ramanujan}'s cubic continued fraction and an analog of his ``{Most Beautiful Identity}''", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "673--680", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003150", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003150", abstract = "In this paper, we prove an analog of Ramanujan's ``Most Beautiful Identity''. This analog is closely related to Ramanujan's beautiful results involving the cubic continued fraction.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chao:2010:NBS, author = "Kuok Fai Chao and Roger Plymen", title = "A new bound for the smallest $x$ with $ \pi (x) > \li (x)$", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "681--690", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003125", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003125", abstract = "We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson [2]. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [exp (727.951858), exp (727.952178)] for which \pi (x) - li(x) > 3.2 $ \times $ 10$^{151}$. There are at least 10$^{154}$ successive integers x in this interval for which \pi (x) > li(x). This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kida:2010:CBQ, author = "Masanari Kida and Y{\=u}ichi Rikuna and Atsushi Sato", title = "Classifying {Brumer}'s Quintic Polynomials by Weak {Mordell--Weil} Groups", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "691--704", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003162", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003162", abstract = "We develop a general classification theory for Brumer's dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lalin:2010:CB, author = "Matilde N. Lal{\'i}n", title = "On a Conjecture by {Boyd}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "3", pages = "705--711", month = may, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003174", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003174", abstract = "The aim of this note is to prove the Mahler measure identity m(x + x$^{-1}$ + y + y$^{-1}$ + 5) = 6m(x + x$^{-1}$ + y + y$^{-1}$ + 1) which was conjectured by Boyd. The proof is achieved by proving relationships between regulators of both curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vulakh:2010:HPM, author = "L. Ya Vulakh", title = "{Hermitian} Points in {Markov} Spectra", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "713--730", month = jun, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110003186", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003186", abstract = "Let H$^n$ be the upper half-space model of the n-dimensional hyperbolic space. For n=3, Hermitian points in the Markov spectrum of the extended Bianchi group B$_d$ are introduced for any d. If \nu is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H$^3$ with diameter 1/\nu, which depends on one continuous parameter. It is shown that \nu$^2$ \leq |D|/24 for any imaginary quadratic field with discriminant D, whose ideal-class group contains no cyclic subgroup of order 4, and in many other cases. Similarly, in the case of n = 4, if \nu is a Hermitian point in the Markov spectrum for SV(Z$^4$), some discrete group of isometries of H$^4$, then the corresponding set of extremal geodesics depends on two continuous parameters.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vulakh:2010:DAI, author = "L. Ya. Vulakh", title = "{Diophantine} Approximation in Imaginary Quadratic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "731--766", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003137", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003137", abstract = "Let H$^3$ be the upper half-space model of the three-dimensional hyperbolic space. For certain cocompact Fuchsian subgroups \Gamma of an extended Bianchi group B$_d$, the extremality of the axis of hyperbolic F \in \Gamma in H$_3$ with respect to \Gamma implies its extremality with respect to B$_d$. This reduction is used to obtain sharp lower bounds for the Hurwitz constants and lower bounds for the highest limit points in the Markov spectra of B$_d$ for some d < 1000. In particular, such bounds are found for all non-Euclidean class one imaginary quadratic fields. The Hurwitz constants for the imaginary quadratic fields with discriminants -120 and -132 are given. The second minima are also indicated for these fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ganguly:2010:DSC, author = "Satadal Ganguly", title = "On the Dimension of the Space of Cusp Forms of Octahedral Type", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "767--783", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003198", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003198", abstract = "For a prime q \equiv 3 (mod 4) and the character, we consider the subspace of the space of holomorphic cusp forms of weight one, level q and character \chi that is spanned by forms that correspond to Galois representations of octahedral type. We prove that this subspace has dimension bounded by upto multiplication by a constant that depends only on \varepsilon.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Rowell:2010:NEL, author = "Michael Rowell", title = "A New Exploration of the {Lebesgue} Identity", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "785--798", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003204", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003204", abstract = "We introduce a new combinatorial proof of the Lebesgue identity which allows us to find a new finite form of the identity. Using this new finite form we are able to make new observations about special cases of the Lebesgue identity, namely the ``little'' G{\"o}llnitz theorems and Sylvester's identity.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lev:2010:ABA, author = "Vsevolod F. Lev and Mikhail E. Muzychuk and Rom Pinchasi", title = "Additive Bases in {Abelian} Groups", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "799--809", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003216", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003216", abstract = "Let G be a finite, non-trivial Abelian group of exponent m, and suppose that B$_1$, \ldots, B$_k$ are generating subsets of G. We prove that if k > 2m ln log$_2$ |G|, then the multiset union B$_1$ \cup B$_k$ forms an additive basis of G; that is, for every g \in G, there exist A$_1$ \subseteq B$_1$, \ldots, A$_k$ \subseteq B$_k$ such that. This generalizes a result of Alon, Linial and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where B$_1$, \ldots, B$_k$ are finite subsets of a vector space, we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form, where A$_i$ vary over all subsets of B$_i$ for each i = 1,\ldots, k. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Volkov:2010:ASS, author = "Maja Volkov", title = "{Abelian} Surfaces with Supersingular Good Reduction and Non-Semisimple {Tate} Module", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "811--818", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003228", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003228", abstract = "We show the existence of abelian surfaces over {$ \mathbb {Q} $}$_p$ having good reduction with supersingular special fiber whose associated $p$-adic Galois module is not semisimple.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2010:RCCb, author = "Hei-Chi Chan", title = "{Ramanujan}'s Cubic Continued Fraction and {Ramanujan} Type Congruences for a Certain Partition Function", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "819--834", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003241", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003241", abstract = "In this paper, we study the divisibility of the function a(n) defined by. In particular, we prove certain ``Ramanujan type congruences'' for a(n) modulo powers of 3.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sinick:2010:RCC, author = "Jonah Sinick", title = "{Ramanujan} Congruences for a Class of Eta Quotients", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "835--847", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003253", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003253", abstract = "We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes \ell for which their coefficients c(n) obey congruences of the form c(\ell n + a) \equiv 0 (mod \ell). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences c$_N$ (n) satisfy, where c$_N$ (n) is defined by. This last result answers a question of H.-C. Chan.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Xia:2010:BNC, author = "Binzhou Xia and Tianxin Cai", title = "{Bernoulli} Numbers and Congruences for Harmonic Sums", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "849--855", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003265", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003265", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Brown:2010:FNH, author = "Jim Brown", title = "The first negative {Hecke} eigenvalue of genus $2$ {Siegel} cuspforms with level $ n \geq 1$", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "857--867", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003277", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003277", abstract = "In this short paper, we extend results of Kohnen and Sengupta on the sign of eigenvalues of Siegel cuspforms. We show that their bound for the first negative Hecke eigenvalue of a genus 2 Siegel cuspform of level 1 extends to the case of level N > 1. We also discuss the signs of Hecke eigenvalues of CAP forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Williams:2010:UIC, author = "Gerald Williams", title = "Unimodular Integer Circulants Associated with Trinomials", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "869--876", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003289", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003289", abstract = "The n $ \times $ n circulant matrix associated with the polynomial (with d < n) is the one with first row (a$_0$ \cdots a$_d$ 0 \cdots 0). The problem as to when such circulants are unimodular arises in the theory of cyclically presented groups and leads to the following question, previously studied by Odoni and Cremona: when is Res(f(t), t$^n$-1) = \pm 1? We give a complete answer to this question for trinomials f(t) = t$^m$ \pm t$^k$ \pm 1. Our main result was conjectured by the author in an earlier paper and (with two exceptions) implies the classification of the finite Cavicchioli--Hegenbarth--Repov{\v{s}} generalized Fibonacci groups, thus giving an almost complete answer to a question of Bardakov and Vesnin.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ahmadi:2010:MOG, author = "Omran Ahmadi and Igor E. Shparlinski and Jos{\'e} Felipe Voloch", title = "Multiplicative Order of {Gauss} Periods", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "877--882", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003290", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003290", abstract = "We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Balazard:2010:CBD, author = "Michel Balazard and Anne {De Roton}", title = "Sur un crit{\`e}re de {B{\'a}ez--Duarte} pour l'hypoth{\`e}se de {Riemann}. ({French}) [{On} a criterion of {B{\'a}ez--Duarte} for the {Riemann Hypothesis}]", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "883--903", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003307", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003307", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Maier:2010:ESP, author = "H. Maier and A. Sankaranarayanan", title = "Exponential Sums Over Primes in Residue Classes", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "905--918", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003319", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003319", abstract = "We specialize a problem studied by Elliott, the behavior of arbitrary sequences a$_p$ of complex numbers on residue classes to prime moduli to the case a$_p$ = e(\alpha p). For these special cases, we obtain under certain additional conditions improvements on Elliott's results.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Roy:2010:SVE, author = "Damien Roy", title = "Small Value Estimates for the Additive Group", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "919--956", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S179304211000323X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000323X", abstract = "We generalize Gel'fond's criterion for algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of $ \mathbb {C} $, instead of just one point (one extension deals with a subgroup of $ \mathbb {C}^\times $).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lagarias:2010:ECS, author = "Jeffrey C. Lagarias", title = "Erratum: {``Cyclic Systems of Simultaneous Congruences''}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "4", pages = "??--??", month = jun, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003320", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:22 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See \cite{Lagarias:2010:CSS}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003320", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jouhet:2010:IEI, author = "Fr{\'e}d{\'e}ric Jouhet and Elie Mosaki", title = "Irrationalit{\'e} aux entiers impairs positifs d'un $q$-analogue de la fonction z{\^e}ta de {Riemann}. ({French}) [{Irrationality} to positive odd integers of a $q$-analogue of the {Riemann} zeta function]", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "959--988", month = aug, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110003332", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003332", abstract = "Dans cet article, nous nous int{\'e}ressons {\`a} un q-analogue aux entiers positifs de la fonction z{\^e}ta de Riemann, que l'on peut {\'e}crire pour s \in \mathbb{N} * sous la forme \zeta$_q$ (s) = $ \sum_{k \geq 1}$ q$^k$ $ \sum_{d|k}$ d$^{s - 1}$. Nous donnons une nouvelle minoration de la dimension de l'espace vectoriel sur {$ \mathbb {Q}$} engendr{\'e}, pour 1/q \in {\mathbb{Z}}{-1; 1} et A entier pair, par 1, \zeta$_q$ (3), \zeta$_q$ (5), \ldots, \zeta$_q$ (A - 1). Ceci am{\'e}liore un r{\'e}sultat r{\'e}cent de Krattenthaler, Rivoal et Zudilin ([13]). En particulier notre r{\'e}sultat a pour cons{\'e}quence le fait que pour 1/q \in {\mathbb{Z}}{-1; 1}, au moins l'un des nombres \zeta$_q$ (3), \zeta$_q$ (5), \zeta$_q$ (7), \zeta$_q$ (9) est irrationnel. In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s \in \mathbb{N} * by \zeta$_q$ (s) = $ \sum_{k \geq 1}$ q$^k$ $ \sum_{d|k}$ d$^{s - 1}$. We give a new lower bound for the dimension of the vector space over {$ \mathbb {Q}$} spanned, for 1/q \in {\mathbb{Z}}{-1; 1} and an even integer A, by 1, \zeta$_q$ (3), \zeta$_q$ (5), \ldots, \zeta$_q$ (A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin ([13]). In particular, a consequence of our result is that for 1/q \in {\mathbb{Z}}{-1; 1}, at least one of the numbers \zeta$_q$ (3), \zeta$_q$ (5), \zeta$_q$ (7), \zeta$_q$ (9) is irrational.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Hasegawa:2010:AOT, author = "Takehiro Hasegawa", title = "On Asymptotically Optimal Towers Over Quadratic Fields Related to {Gauss} Hypergeometric Functions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "989--1009", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003344", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003344", abstract = "We define two asymptotically optimal towers over quadratic fields, and give the explicit descriptions of the ramification loci and the sets of places splitting completely, which relate to the elliptic modular curves X$_0$ (4$^n$) and X$_0$ (3$^n$ ), respectively. Moreover, in the last section, we determine completely the modularity of a tower given by Maharaj and Wulftange in [18].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ih:2010:FPP, author = "Su-Ion Ih and Thomas J. Tucker", title = "A Finiteness Property for Preperiodic Points of {Chebyshev} Polynomials", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1011--1025", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003356", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003356", abstract = "Let K be a number field with algebraic closure, let S be a finite set of places of K containing the Archimedean places, and let \phi be a Chebyshev polynomial. We prove that if is not preperiodic, then there are only finitely many preperiodic points which are S-integral with respect to \alpha.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gottesman:2010:QRP, author = "Richard Gottesman and Kwokfung Tang", title = "Quadratic Recurrences with a Positive Density of Prime Divisors", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1027--1045", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003368", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003368", abstract = "For f(x) \in {\mathbb{Z}}[x] and a \in {\mathbb{Z}}, we let f$^n$ (x) be the nth iterate of f(x), P(f, a) = {p prime: p|f$^n$ (a) for some n}, and D(P(f, a)) denote the natural density of P(f, a) within the set of primes. A conjecture of Jones [5] indicates that D(P(f, a)) = 0 for most quadratic f. In this paper, we find an exceptional family of (f, a) such that D(P(f, a)) > 0 by considering f$_t$ (x) = (x + t)$^2$- 2 - t and a$_t$ = f$_t$ (0) for t \in {\mathbb{Z}}. We prove that if t is not of the form \pm M$^2$ \pm 2 or \pm 2M$^2$ \pm 2, then D(P(f$_t$, a$_t$)) = {\u{2}153}. We also determine D(P(f$_t$, a$_t$)) in some cases when the density is not equal to {\u{2}153}. Our results suggest a connection between the arithmetic dynamics of the conjugates of x$^2$ and the conjugates of x$^2$- 2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hoshi:2010:FIP, author = "Akinari Hoshi and Katsuya Miyake", title = "On the Field Intersection Problem of Solvable Quintic Generic Polynomials", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1047--1081", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S179304211000337X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000337X", abstract = "We study a general method of the field intersection problem of generic polynomials over an arbitrary field k via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using multi-resolvent polynomials.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Knopp:2010:ECG, author = "Marvin Knopp and Wissam Raji", title = "{Eichler} Cohomology for Generalized Modular Forms {II}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1083--1090", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S179304211000340X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000340X", abstract = "We derive further results on Eichler cohomology of generalized modular forms of arbitrary real weight.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Agashe:2010:SSV, author = "Amod Agashe", title = "Squareness in the Special {$L$}-Value and Special {$L$}-Values of Twists", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1091--1111", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003393", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003393", abstract = "Let N be a prime and let A be a quotient of J$_0$ (N) over Q associated to a newform such that the special $L$-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special $L$-value of A is divisible by an odd prime q such that q does not divide the numerator of. Then the Birch and Swinnerton-Dyer conjecture predicts that the $q$-adic valuations of the algebraic part of the special $L$-value of A and of the order of the Shafarevich--Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special $L$-values of twists of A, we show that the $q$-adic valuations of the algebraic part of the special $L$-value of A and of the Birch and Swinnerton-Dyer conjectural order of the Shafarevich--Tate group of A are both positive even numbers. We also give a formula for the algebraic part of the special $L$-value of A over quadratic imaginary fields K in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently, over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and \infty) which shows that this algebraic part is a perfect square up to powers of the prime two and of primes dividing the discriminant of K. Finally, for an optimal elliptic curve of arbitrary conductor E, we give a formula for the special $L$-value of the twist E$_{-D}$ of E by a negative fundamental discriminant -D, which shows that this special $L$-value is an integer up to a power of 2, under some hypotheses. In view of the second part of the Birch and Swinnerton-Dyer conjecture, this leads us to the surprising conjecture that the square of the order of the torsion subgroup of E$_{-D}$ divides the product of the order of the Shafarevich--Tate group of E$_{-D}$ and the orders of the arithmetic component groups of E$_{-D}$, under certain mild hypotheses.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Matomaki:2010:NSN, author = "Kaisa Matom{\"a}ki", title = "A Note on Smooth Numbers in Short Intervals", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1113--1116", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003381", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003381", abstract = "We prove that, for any \in > 0, there exists a constant C > 0 such that the interval contains numbers whose all prime factors are smaller than.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Shemanske:2010:CSH, author = "T. Shemanske and S. Treneer and L. Walling", title = "Constructing Simultaneous {Hecke} Eigenforms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1117--1137", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003411", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003411", abstract = "It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie--Kohnen who considered diagonalization of ``bad'' Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kleinbock:2010:MDA, author = "Dmitry Kleinbock and Gregory Margulis and Junbo Wang", title = "Metric {Diophantine} Approximation for Systems of Linear Forms Via Dynamics", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1139--1168", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003423", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003423", abstract = "The goal of this paper is to generalize the main results of [21] and subsequent papers on metric with dependent quantities to the set-up of systems of linear forms. In particular, we establish ``joint strong extremality'' of arbitrary finite collection of smooth non-degenerate submanifolds of {\mathbb{R}}$^n$. The proofs are based on generalized quantitative non-divergence estimates for translates of measures on the space of lattices.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hoelscher:2010:RCG, author = "Jing Long Hoelscher", title = "Ray Class Groups of Quadratic and Cyclotomic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1169--1182", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003447", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003447", abstract = "This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Let be a real quadratic number field with a prime P lying above p in {$ \mathbb {Q} $}. If p splits in K/{$ \mathbb {Q} $} and p does not divide the big class number of K, then any pro-p extension of K ramified only at P is finite cyclic. If p is inert in K/{$ \mathbb {Q} $}, then there exist infinite extensions of K ramified only at P. Furthermore, for big enough integer k, the ray class field (mod P$^{k + 1}$) is obtained from the ray class field (mod P$^k$) by adjoining $ \zeta_{p^{k + 1}}$. In the case of a regular cyclotomic number field $ K = \mathbb {Q}(\zeta_p)$, the explicit structure of ray class groups $ (m o d P^k)$ is given for any positive integer $k$, where $P$ is the unique prime in $K$ above $p$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ulas:2010:VHT, author = "Maciej Ulas", title = "Variations on Higher Twists of Pairs of Elliptic Curves", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1183--1189", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003472", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003472", abstract = "In this note we show that for any pair of elliptic curves E$_1$, E$_2$ over {$ \mathbb {Q}$} with j-invariant equal to 0, we can find a polynomial D \in {\mathbb{Z}}[u, v, w, t] such that the sextic twists of the curves E$_1$, E$_2$ by D(u, v, w, t) have rank \geq 2 over the field {$ \mathbb {Q}$}(u, v, w, t). A similar result is proved for simultaneous quartic twists of pairs of elliptic curves with j-invariant 1728.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Villa-Salvador:2010:EPC, author = "Gabriel Villa-Salvador", title = "An Elementary Proof of the Conductor--Discriminant Formula", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "5", pages = "1191--1197", month = aug, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003459", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003459", abstract = "For a finite abelian extension K/{$ \mathbb {Q} $}, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the $L$-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of p$^n$-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baczkowski:2010:V, author = "Daniel Baczkowski and Michael Filaseta and Florian Luca and Ognian Trifonov", title = "On values of $ d(n!) / m! $, $ \varphi (n!) / m! $ and $ \sigma (n!) / m! $", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1199--1214", month = sep, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110003435", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003435", abstract = "For f one of the classical arithmetic functions d, \varphi and \sigma, we establish constraints on the quadruples (n, m, a, b) of integers satisfying f(n!)/m! = a/b. In particular, our results imply that as nm tends to infinity, the number of distinct prime divisors dividing the product of the numerator and denominator of the fraction f(n!)/m!, when reduced, tends to infinity.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kable:2010:AIO, author = "Anthony C. Kable", title = "An Arithmetical Invariant of Orbits of Affine Actions and Its Application to Similarity Classes of Quadratic Spaces", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1215--1253", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003460", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003460", abstract = "Given an action of an affine algebraic group on an affine variety and a relatively invariant regular function, all defined over the ring of integers of a number field and having suitable additional properties, an invariant of the rational orbits of the action is defined. This invariant, the reduced replete Steinitz class, takes its values in the reduced replete class group of the number field. The general framework is then applied to obtain an invariant of similarity classes of non-degenerate quadratic spaces of even rank. The invariant is related to more familiar invariants. It is shown that if the similarity classes are weighted by the volume of an associated automorphism group then their reduced replete Steinitz classes are asymptotically uniformly distributed with respect to a natural parameter.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kohnen:2010:SNF, author = "Winfried Kohnen", title = "A Short Note on {Fourier} Coefficients of Half-Integral Weight Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1255--1259", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003484", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003484", abstract = "We give an unconditional proof of a result on sign changes of Fourier coefficients of cusp forms of half-integral weight that before was proved only under the hypothesis of Chowla's conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vaserstein:2010:PPP, author = "Leonid Vaserstein and Takis Sakkalis and Sophie Frisch", title = "Polynomial Parametrization of {Pythagorean} Tuples", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1261--1272", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003496", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003496", abstract = "A Pythagorean (k, l)-tuple over a commutative ring A is a vector x = (x$_i$) \in A$^{k + l}$, where k, l \in \mathbb{N}, k \geq l which satisfies. In this paper, a polynomial parametrization of Pythagorean (k, l)-tuples over the ring F[t] is given, for l \geq 2. In the case where l = 1, solutions of the above equation are provided for k = 2, 3, 4, 5, and 9.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mammo:2010:DDA, author = "Behailu Mammo", title = "On the Density of Discriminants of {Abelian} Extensions of a Number Field", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1273--1291", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003502", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003502", abstract = "Let G = C$_{\ell }$ $ \times $ C$_{\ell }$ denote the product of two cyclic groups of prime order \ell, and let k be an algebraic number field. Let N(k, G, m) denote the number of abelian extensions K of k with Galois group G(K/k) isomorphic to G, and the relative discriminant {$ \mathcal {D} $}(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for $ \sum_{m \leq X}$ N(k, G; m). This extends the result previously obtained by Datskovsky and Mammo.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lozano-Robledo:2010:BTE, author = "{\'A}lvaro Lozano-Robledo and Benjamin Lundell", title = "Bounds for the Torsion of Elliptic Curves Over Extensions with Bounded Ramification", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1293--1309", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003514", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003514", abstract = "Let E be a semi-stable elliptic curve defined over {$ \mathbb {Q} $}, and fix N \geq 2. Let $ K_N $ /{$ \mathbb {Q} $} be a maximal algebraic Galois extension of {$ \mathbb {Q} $} whose ramification indices are all at most N. We show that there exists a computable bound B(N), which depends only on N and not on the choice of E/{$ \mathbb {Q} $}, such that the size of E(K$_N$)$_{Tors}$ is always at most B(N).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jahangiri:2010:GAQ, author = "Majid Jahangiri", title = "Generators of Arithmetic Quaternion Groups and a {Diophantine} Problem", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1311--1328", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003551", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003551", abstract = "Let p be a prime and a a quadratic non-residue (mod p). Then the set of integral solutions of the Diophantine equation form a cocompact discrete subgroup \Gamma$_{p, a}$ \subset SL(2, {\mathbb{R}}) which is commensurable with the group of units of an order in a quaternion algebra over {$ \mathbb {Q}$}. The problem addressed in this paper is an estimate for the traces of a set of generators for \Gamma$_{p, a}$. Empirical results summarized in several tables show that the trace has significant and irregular fluctuations which is reminiscent of the behavior of the size of a generator for the solutions of Pell's equation. The geometry and arithmetic of the group of units of an order in a quaternion algebra play a key role in the development of the code for the purpose of this paper.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Tanti:2010:ECS, author = "Jagmohan Tanti and S. A. Katre", title = "{Euler}'s Criterion for Septic Nonresidues", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1329--1347", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003563", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003563", abstract = "Let p be a prime \equiv 1 (mod 7). In this paper, we obtain an explicit expression for a primitive seventh root of unity (mod p) in terms of coefficients of a Jacobi sum of order 7 and also in terms of a solution of a Diophantine system of Leonard and Williams, and then obtain Euler's criterion for septic nonresidues D (mod p) in terms of this seventh root. Explicit results are given for septic nonresidues for D = 2, 3, 5, 7.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Booher:2010:ECT, author = "Jeremy Booher and Anastassia Etropolski and Amanda Hittson", title = "Evaluations of Cubic Twisted {Kloosterman} Sheaf Sums", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1349--1365", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003538", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003538", abstract = "We prove some conjectures of Evans and Katz presented in a paper by Evans regarding twisted Kloosterman sheaf sums T$_n$. These conjectures give explicit evaluations of the sums T$_n$ where the character is cubic and n = 4. There are also conjectured relationships between evaluations of T$_n$ and the coefficients of certain modular forms. For three of these modular forms, each of weight 3, it is conjectured that the coefficients are related to the squares of the coefficients of weight 2 modular forms. We prove these relationships using the theory of complex multiplication.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Verrill:2010:CRM, author = "H. A. Verrill", title = "Congruences Related to Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1367--1390", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003587", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003587", abstract = "Let $f$ be a modular form of weight $k$ for a congruence subgroup $ \Gamma \subset \mathrm {SL}_2 (Z)$, and $t$ a weight $0$ modular function for $ \Gamma $. Assume that near $ t = 0$, we can write $ f = \sum_{n \geq 0} b_n t^n$, $ b_n \in Z$. Let $ \ell (z)$ be a weight $ k + 2$ modular form with $q$-expansion $ \sum \gamma_n q^n$, such that the Mellin transform of $ \ell $ can be expressed as an Euler product. Then we show that if for some integers $ a_i$, $ d_i$, then the congruence relation $ b_{mp^r} - \gamma_p b_{mp^{r - 1}} + \varepsilon_p p^{k + 1} b_{mp^{r - 2}} \equiv 0 (\bmod p^r)$ holds. We give a number of examples of this phenomena.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{David:2010:SFD, author = "Chantal David and Jorge Jim{\'e}nez Urroz", title = "Square-Free Discriminants of {Frobenius} Rings", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1391--1412", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003599", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003599", abstract = "Let E be an elliptic curve over {$ \mathbb {Q} $}. We know that the ring of endomorphisms of its reduction modulo an ordinary prime p is an order of the quadratic imaginary field generated by the Frobenius element \pi$_p$. However, except in the trivial case of complex multiplication, very little is known about the fields that appear as algebras of endomorphisms when p varies. In this paper, we study the endomorphism ring by looking at the arithmetic of, the discriminant of the characteristic polynomial of \pi$_p$. In particular, we give a precise asymptotic for the function counting the number of primes p up to x such that is square-free and in certain congruence class fixed {\em a priori\/}, when averaging over elliptic curves defined over the rationals. We discuss the relation of this result with the Lang--Trotter conjecture, and some other questions on the curve modulo p.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hambleton:2010:QRR, author = "S. Hambleton and V. Scharaschkin", title = "Quadratic Reciprocity Via Resultants", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1413--1417", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S179304211000354X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000354X", abstract = "We give a simple inductive proof of quadratic reciprocity using resultants.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lebacque:2010:TVI, author = "Philippe Lebacque", title = "On {Tsfasman--Vl{\u{a}}du{\c{t}}} invariants of infinite global fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "6", pages = "1419--1448", month = sep, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003526", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003526", abstract = "In this paper, we study certain asymptotic properties of global fields. We consider the set of Tsfasman--Vl{\u{a}}du{\c{t}} invariants of infinite global fields and answer some natural questions arising from their work. In particular, we prove the existence of infinite global fields having finitely many strictly positive invariants at given places, and the existence of infinite number fields with certain prescribed invariants being zero. We also give precisions on the deficiency of infinite global fields and on the primes decomposition in those fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dujella:2010:SSP, author = "Andrej Dujella and Ana Jurasi{\'c}", title = "On the Size of Sets in a Polynomial Variant of a Problem of {Diophantus}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1449--1471", month = nov, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110003575", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003575", abstract = "In this paper, we prove that there does not exist a set of 8 polynomials (not all constant) with coefficients in an algebraically closed field of characteristic 0 with the property that the product of any two of its distinct elements plus 1 is a perfect square.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ehlen:2010:TBP, author = "Stephan Ehlen", title = "Twisted {Borcherds} Products on {Hilbert} Modular Surfaces and the Regularized Theta Lift", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1473--1489", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003642", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003642", abstract = "We construct a lifting from weakly holomorphic modular forms of weight 0 for SL$_2$ ({\mathbb{Z}}) with integral Fourier coefficients to meromorphic Hilbert modular forms of weight 0 for the full Hilbert modular group of a real quadratic number field with an infinite product expansion and a divisor given by a linear combination of twisted Hirzebruch--Zagier divisors. The construction uses the singular theta lifting by considering a suitable twist of a Siegel theta function. We generalize the work by Bruinier and Yang who showed the existence of the lifting for prime discriminants using a different approach.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Belabas:2010:DCP, author = "Karim Belabas and {\'E}tienne Fouvry", title = "Discriminants cubiques et progressions arithm{\'e}tiques. ({French}) [{Cubic} discriminants and arithmetic progressions]", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1491--1529", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003605", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003605", abstract = "Nous calculons la densit{\'e} des discriminants des corps sextiques galoisiens de groupe S$_3$, d{\'e}montrant un nouveau cas de la conjecture de Malle ainsi qu'un cas particulier de sa g{\'e}n{\'e}ralisation par Ellenberg et Venkatesh. Plus g{\'e}n{\'e}ralement, nous {\'e}tudions la densit{\'e} des discriminants de corps cubiques dans une progression arithm{\'e}tique, avec une zone d'uniformit{\'e} la plus large possible. We compute the density of discriminants of Galois sextic fields with group S$_3$, thereby proving a new case of Malle's conjecture as well as a special case of its generalization by Ellenberg and Venkatesh. Further, we study the density of cubic discriminants in an arithmetic progression, in the largest possible uniformity with respect to the modulus.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Patkowski:2010:CGF, author = "Alexander E. Patkowski", title = "On Curious Generating Functions for Values of {$L$}-Functions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1531--1540", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003630", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003630", abstract = "We prove some curious identities for generating functions for values of {$L$}-functions. It is shown how to obtain generating functions for values of {$L$}-functions using a slightly different approach, resulting in some new $q$-series identities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Wu:2010:RGD, author = "Qingquan Wu and Renate Scheidler", title = "The Ramification Groups and Different of a Compositum of {Artin--Schreier} Extensions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1541--1564", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003617", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003617", abstract = "Let K be a function field over a perfect constant field of positive characteristic p, and L the compositum of n (degree p) Artin--Schreier extensions of K. Then much of the behavior of the degree p$^n$ extension L/K is determined by the behavior of the degree p intermediate extensions M/K. For example, we prove that a place of K totally ramifies/is inert/splits completely in L if and only if it totally ramifies/is inert/splits completely in every M. Examples are provided to show that all possible decompositions are in fact possible; in particular, a place can be inert in a non-cyclic Galois function field extension, which is impossible in the case of a number field. Moreover, we give an explicit closed form description of all the different exponents in L/K in terms of those in all the M/K. Results of a similar nature are given for the genus, the regulator, the ideal class number and the divisor class number. In addition, for the case n = 2, we provide an explicit description of the ramification group filtration of L/K.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pickett:2010:CSD, author = "Erik Jarl Pickett", title = "Construction of Self-Dual Integral Normal Bases in {Abelian} Extensions of Finite and Local Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1565--1588", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003654", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003654", abstract = "Let F/E be a finite Galois extension of fields with abelian Galois group \Gamma. A self-dual normal basis for F/E is a normal basis with the additional property that Tr$_{F / E}$ (g(x), h(x)) = \delta$_{g, h}$ for g, h \in \Gamma. Bayer-Fluckiger and Lenstra have shown that when char(E) \neq 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char(E) = 2, then F admits a self-dual normal basis if and only if the exponent of \Gamma is not divisible by 4. In this paper, we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let K be a finite extension of {$ \mathbb {Q}$}$_p$, let L/K be a finite abelian Galois extension of odd degree and let be the valuation ring of L. We define A$_{L / K}$ to be the unique fractional -ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for A$_{L / K}$ if and only if L/K is weakly ramified. Assuming p \neq 2, we construct such bases whenever they exist.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Robertson:2010:MCF, author = "Leanne Robertson", title = "Monogeneity in Cyclotomic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1589--1607", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003666", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003666", abstract = "A number field is said to be {\em monogenic\/} if its ring of integers is a simple ring extension {\mathbb{Z}}[\alpha ] of {\mathbb{Z}}. It is a classical and usually difficult problem to determine whether a given number field is monogenic and, if it is, to find all numbers \alpha that generate a power integral basis {1, \alpha, \alpha$^2$, \ldots, \alpha$^k$ } for the ring. The nth cyclotomic field {$ \mathbb {Q}$}(\zeta$_n$) is known to be monogenic for all n, and recently Ranieri proved that if n is coprime to 6, then up to integer translation all the integral generators for {$ \mathbb {Q}$}(\zeta$_n$) lie on the unit circle or the line Re(z) = 1/2 in the complex plane. We prove that this geometric restriction extends to the cases n = 3k and n = 4k, where k is coprime to 6. We use this result to find all power integral bases for {$ \mathbb {Q}$}(\zeta$_n$) for n = 15, 20, 21, 28. This leads us to a conjectural solution to the problem of finding all integral generators for cyclotomic fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Feigon:2010:EAC, author = "Brooke Feigon and David Whitehouse", title = "Exact Averages of Central Values of Triple Product {$L$}-Functions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1609--1624", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S179304211000368X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000368X", abstract = "We obtain exact formulas for central values of triple product {$L$}-functions averaged over newforms of weight 2 and prime level. We apply these formulas to non-vanishing problems. This paper uses a period formula for the triple product {$L$}-function proved by Gross and Kudla.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Katayama:2010:GGF, author = "Koji Katayama", title = "Generalized Gamma Functions with Characters", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1625--1657", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003629", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003629", abstract = "The main objective of this paper is to define \Gamma functions \Gamma [\chi ](v) with characters \chi and study their properties. To this end, we ought to introduce {$L$}-functions of Hurwitz type. We prove that holds, which combines the theory at ``s = 0'' and the theory at ``s = 1''.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bostan:2010:GFC, author = "Alin Bostan and Bruno Salvy and Khang Tran", title = "Generating Functions of {Chebyshev}-Like Polynomials", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1659--1667", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003691", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003691", abstract = "In this short note, we give simple proofs of several results and conjectures formulated by Stolarsky and Tran concerning generating functions of some families of Chebyshev-like polynomials.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lee:2010:SPF, author = "K. S. Enoch Lee", title = "On the Sum of a Prime and a {Fibonacci} Number", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1669--1676", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003708", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003708", abstract = "We show that the set of the numbers that are the sum of a prime and a Fibonacci number has positive lower asymptotic density.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dewar:2010:RCS, author = "Michael Dewar and Olav K. Richter", title = "{Ramanujan} Congruences for {Siegel} Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1677--1687", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S179304211000371X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000371X", abstract = "We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Schwab:2010:GAF, author = "Emil Daniel Schwab", title = "Generalized Arithmetical Functions of Three Variables", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1689--1699", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003721", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003721", abstract = "The paper is devoted to the study of some properties of generalized arithmetical functions extended to the case of three variables. The convolution in this case is a convolution of the incidence algebra of a M{\"o}bius category in the sense of Leroux. This category is a two-sided analogue of the poset (it is viewed as a category) of positive integers ordered by divisibility.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sairaiji:2010:FGJ, author = "Fumio Sairaiji", title = "Formal Groups of {Jacobian} Varieties of Hyperelliptic Curves", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "7", pages = "1701--1716", month = nov, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003733", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:23 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003733", abstract = "Let k be a field of characteristic zero. In this paper, we discuss two explicit constructions of the formal groups {\u{0}134} of the Jacobian varieties J of hyperelliptic curves C over k. Our results are generalizations of the classical constructions of formal groups of elliptic curves. As an application of our results, we may decide the type of bad reduction of J modulo p when C is a hyperelliptic curve over {$ \mathbb {Q} $}.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Templier:2010:AVC, author = "Nicolas Templier", title = "On Asymptotic Values of Canonical Quadratic {$L$}-Functions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1717--1730", month = dec, year = "2010", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042110003678", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003678", abstract = "We establish an asymptotic for the first moment of Hecke $L$-series associated to canonical characters on imaginary quadratic fields. This provides another proof and improves recent results by Masri and Kim--Masri--Yang.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Samuels:2010:FCU, author = "Charles L. Samuels", title = "The Finiteness of Computing the Ultrametric {Mahler} Measure", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1731--1753", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003745", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003745", abstract = "Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted M$_{\infty }$ (\alpha) for an algebraic number \alpha. We show that the computation of M$_{\infty }$ (\alpha) can be reduced to a certain search through a finite set. Although it is an open problem to record the points of this set in general, we provide some examples where it is reasonable to compute and our result can be used to determine M$_{\infty }$ (\alpha).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kang:2010:APT, author = "Soon-Yi Kang and Chang Heon Kim", title = "Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1755--1768", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003757", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003757", abstract = "After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sbeity:2010:CSE, author = "Farah Sbeity and Boucha{\"i}b Soda{\"i}gui", title = "Classes de {Steinitz} d'extensions non ab{\'e}liennes {\`a} groupe de {Galois} d'ordre $1$6 ou extrasp{\'e}cial d'ordre $ 32$ et probl{\`e}me de plongement. ({French}) [{Steinitz} classes of non-Abelian extensions to {Galois} group of order $ 16 $ or extraspecial of order $ 32$ and embedding problem]", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1769--1783", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003794", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003794", abstract = "Soient k un corps de nombres et Cl(k) son groupe des classes. Soit \Gamma un groupe non ab{\'e}lien d'ordre 16, ou un groupe extrasp{\'e}cial d'ordre 32. Soit R$_m$ (k, \Gamma) le sous-ensemble de Cl(k) form{\'e} par les {\'e}l{\'e}ments qui sont r{\'e}alisables par les classes de Steinitz d'extensions galoisiennes de k, mod{\'e}r{\'e}ment ramifi{\'e}es et dont le groupe de Galois est isomorphe {\`a} \Gamma. Lorsque \Gamma est le groupe modulaire d'ordre 16, on suppose que k contienne une racine primitive 4{\`e}me de l'unit{\'e}. Dans cet article on montre que R$_m$ (k, \Gamma) est le groupe Cl(k) tout entier si le nombre des classes de k est impair. On {\'e}tudie un probl{\`e}me de plongement en liaison avec les classes de Steinitz dans la perspective de l'{\'e}tude des classes galoisiennes r{\'e}alisables. On prouve que pour tout c \in Cl(k), il existe une extension quadratique de k, mod{\'e}r{\'e}e, dont la classe de Steinitz est c, et qui est plongeable dans une extension galoisienne de k, mod{\'e}r{\'e}e et {\`a} groupe de Galois isomorphe {\`a} \Gamma. Let k be a number field and Cl(k) its class group. Let \Gamma be a nonabelian group of order 16 or an extra-special group of order 32. Let R$_m$ (k, \Gamma) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to \Gamma. When \Gamma is the modular group of order 16, we assume that k contains a primitive 4th root of unity. In the present paper, we show that R$_m$ (k, \Gamma) is the full group Cl(k) if the class number of k is odd. We study an embedding problem connected with Steinitz classes in the perspective of studying realizable Galois module classes. We prove that for all c \in Cl(k), there exist a tame quadratic extension of k, with Steinitz class c, and which is embeddable in a tame Galois extension of k with Galois group isomorphic to \Gamma.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Lucht:2010:SRE, author = "Lutz G. Lucht", title = "A Survey of {Ramanujan} Expansions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1785--1799", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003800", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003800", abstract = "This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujan's paper in 1918, following Carmichael's mean-value-based concept from 1932 up to 1994. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on Ramanujan expansions. In particular, very short proofs of theorems for additive and multiplicative functions going far beyond previous borders are presented, and Ramanujan expansions that formerly have been considered mysterious are explained.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cai:2010:LFS, author = "Yingchun Cai", title = "{Lagrange}'s Four Squares Theorem with Variables of Special Type", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1801--1817", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003812", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003812", abstract = "Let N denote a sufficiently large integer satisfying N \equiv 4 (mod 24), and P$_r$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that the equation is solvable in one prime and three P$_{42}$, or in four P$_{13}$. These results constitute improvements upon that of Heath-Brown and Tolev.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Barman:2010:IIT, author = "Rupam Barman and Anupam Saikia", title = "{Iwasawa} $ \lambda $-invariants and {$ \Gamma $}-transforms of $p$-adic measures on {$ \mathbb {Z}_p^n $}", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1819--1829", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003824", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003824", abstract = "In this paper, we determine a relation between the \lambda -invariants of a $p$-adic measure on and its \Gamma transform. Along the way we also determine $p$-adic properties of certain Mahler coefficients.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Caranay:2010:ESP, author = "Perlas C. Caranay and Renate Scheidler", title = "An Efficient Seventh Power Residue Symbol Algorithm", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1831--1853", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003770", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003770", abstract = "Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In this paper, we provide an efficient procedure for computing 7th power residue symbols. The method employs arithmetic in the field {$ \mathbb {Q} $}(\zeta), with \zeta a primitive 7th root of unity, and its ring of integers {\mathbb{Z}}[\zeta ]. We give an explicit characterization for an element in {\mathbb{Z}}[\zeta ] to be primary, and provide an algorithm for finding primary associates of integers in {\mathbb{Z}}[\zeta ]. Moreover, we formulate explicit forms of the complementary laws to Kummer's 7th degree reciprocity law, and use Lenstra's norm-Euclidean algorithm in the cyclotomic field.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{DelCorso:2010:NII, author = "Ilaria {Del Corso} and Roberto Dvornicich", title = "Non-Invariance of the Index in Wildly Ramified Extensions", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1855--1868", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003836", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003836", abstract = "In this paper, we give an example of three wildly ramified extensions L$_1$, L$_2$, L$_3$ of {$ \mathbb {Q}$}$_2$ with the same ramification numbers and isomorphic Galois groups, such that I(nL$_1$ ) > I(nL$_2$) > I(nL$_3$) for a suitable integer n (where I(nL) denotes the index of the {$ \mathbb {Q}$}$_2$-algebra L$^n$). This example shows that the condition given in [2] for the invariance of the index of tamely ramified extensions is no longer sufficient in the general case.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Laishram:2010:CRP, author = "Shanta Laishram", title = "On a Conjecture on {Ramanujan} Primes", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1869--1873", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003848", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003848", abstract = "For n \geq 1, the {\em nth Ramanujan prime\/} is defined to be the smallest positive integer R$_n$ with the property that if x \geq R$_n$, then where \pi (\nu) is the number of primes not exceeding \nu for any \nu > 0 and \nu \in {\mathbb{R}}. In this paper, we prove a conjecture of Sondow on upper bound for Ramanujan primes. An explicit bound of Ramanujan primes is also given. The proof uses explicit bounds of prime \pi and \theta functions due to Dusart.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vienney:2010:NCA, author = "Mathieu Vienney", title = "A new construction of $p$-adic {Rankin} convolutions in the case of positive slope", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1875--1900", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003782", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003782", abstract = "Given two newforms f and g of respective weights k and l with l < k, Hida constructed a $p$-adic {$L$}-function interpolating the values of the Rankin convolution of f and g in the critical strip l \leq s \leq k. However, this construction works only if f is an ordinary form. Using a method developed by Panchishkin to construct $p$-adic {$L$}-function associated with modular forms, we generalize this construction to the case where the slope of f is small.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Brown:2010:SVF, author = "Jim Brown", title = "Special values of {$L$}-functions on {$ \mathrm {GSp}_4 \times \mathrm {GL}_2$} and the non-vanishing of {Selmer} groups", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1901--1926", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003769", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003769", abstract = "In this paper, we show how one can use an inner product formula of Heim giving the inner product of the pullback of an Eisenstein series from Sp$_{10}$ to Sp$_2$ $ \times $ Sp$_4$ $ \times $ Sp$_4$ with a new-form on GL$_2$ and a Saito--Kurokawa lift to produce congruences between Saito--Kurokawa lifts and non-CAP forms. This congruence is in part controlled by the {$L$}-function on GSp$_4$ $ \times $ GL$_2$. The congruence is then used to produce nontrivial torsion elements in an appropriate Selmer group, providing evidence for the Bloch--Kato conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kaneko:2010:KDM, author = "Masanobu Kaneko and Yasuo Ohno", title = "On a Kind of Duality of Multiple Zeta-Star Values", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1927--1932", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S179304211000385X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211000385X", abstract = "A duality-type relation for height one multiple zeta-star values is established. A conjectural generalization to the case of arbitrary height is also presented.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bettin:2010:SMR, author = "Sandro Bettin", title = "The Second Moment of the {Riemann} Zeta Function with Unbounded Shifts", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1933--1944", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003861", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003861", abstract = "We prove an asymptotic formula for the second moment (up to height T) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to T$^{2 - \varepsilon }$, for any \varepsilon > 0.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Anonymous:2010:AIV, author = "Anonymous", title = "Author Index (Volume 6)", journal = j-INT-J-NUMBER-THEORY, volume = "6", number = "8", pages = "1945--1950", month = dec, year = "2010", DOI = "https://doi.org/10.1142/S1793042110003885", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042110003885", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Joshi:2011:IHC, author = "Kirti Joshi and Cameron Mcleman", title = "Infinite {Hilbert} Class Field Towers from {Galois} Representations", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "1--8", month = feb, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042111003879", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003879", abstract = "We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each k \in {12, 16, 18, 20, 22, 26}, we give explicit rational primes \ell such that the fixed field of the mod-\ell representation attached to the unique normalized cusp eigenform of weight k on SL$_2$ ({\mathbb{Z}}) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes \ell for each k in the list. Finally, given a non-CM curve E/{$ \mathbb {Q}$}, we show that there exists an integer M$_E$ such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to M$_E$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Nguyen:2011:CDM, author = "Lan Nguyen", title = "A Complete Description of Maximal Solutions of Functional Equations Arising from Multiplication of Quantum Integers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "9--56", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003909", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003909", abstract = "In this paper, we resolve a problem raised by Nathanson concerning the maximal solutions to functional equations naturally arising from multiplication of quantum integers ([4]). Together with our results obtained in [11], which treats the case where the field of coefficients is {$ \mathbb {Q} $}, this provides a complete description of the maximal solutions to these functional equations and their support bases P in characteristic zero setting.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Heuberger:2011:PDA, author = "Clemens Heuberger and Helmut Prodinger", title = "A precise description of the $p$-adic valuation of the number of alternating sign matrices", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "57--69", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003892", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003892", abstract = "Following Sun and Moll ([4]), we study v$_p$ (T(N)), the $p$-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behavior, by means of Fourier coefficients. The method is the Mellin--Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zhang:2011:FPM, author = "Deyu Zhang and Wenguang Zhai", title = "On the fifth-power moment of {$ \Delta (x) $}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "71--86", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003922", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003922", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Buckingham:2011:FGI, author = "Paul Buckingham", title = "The Fractional {Galois} Ideal for Arbitrary Order of Vanishing", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "87--99", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004010", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004010", abstract = "We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those {$L$}-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include {$L$}-functions with a simple zero at s = 0, and was shown by the present author to provide class-Group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with {$L$}-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to B{\"u}y{\"u}kboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gurak:2011:GKS, author = "S. Gurak", title = "{Gauss} and {Kloosterman} Sums Over Residue Rings of Algebraic Integers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "101--114", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003958", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003958", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bremner:2011:RPS, author = "Andrew Bremner and Maciej Ulas", title = "Rational Points on Some Hyper- and Superelliptic Curves", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "115--132", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003946", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003946", abstract = "We construct families of certain hyper- and superelliptic curves that contain a (small) number of rational points. This leads to lower bounds for the ranks of Jacobians of certain high genus curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Fu:2011:CPO, author = "Shishuo Fu", title = "Combinatorial Proof of One Congruence for the Broken $1$-Diamond Partition and a Generalization", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "133--144", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004022", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004022", abstract = "In one of their recent collaborative papers, Andrews and Paule continue their study of partition functions via MacMahon's Partition Analysis by considering partition functions associated with directed graphs which consist of chains of diamond shape. They prove a congruence related to one of these partition functions and conjecture a number of similar congruence results. In this note, we reprove this congruence by constructing an explicit way to group partitions. Then we keep the essence of the method and manage to apply it to a different kind of plane partitions to get more general results and several other congruences.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cosgrave:2011:MOC, author = "John B. Cosgrave and Karl Dilcher", title = "The Multiplicative Orders of Certain {Gauss} Factorials", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "145--171", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S179304211100396X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100396X", abstract = "A theorem of Gauss extending Wilson's theorem states the congruence (n - 1)$_n$ ! \equiv -1 (mod n) whenever n has a primitive root, and \equiv 1 (mod n) otherwise, where N$_n$ ! denotes the product of all integers up to N that are relatively prime to n. In the spirit of this theorem, we study the multiplicative orders of (mod n) for odd prime powers p$^{\alpha }$. We prove a general result about the connection between the order for p$^{\alpha }$ and for p$^{\alpha + 1}$ and study exceptions to the general rule. Particular emphasis is given to the cases M = 3, M = 4 and M = 6, while the case M = 2 is already known.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Carls:2011:GTC, author = "Robert Carls", title = "{Galois} Theory of the Canonical Theta Structure", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "173--202", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003934", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/agm.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003934", abstract = "In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain $p$-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hubrechts:2011:MEH, author = "Hendrik Hubrechts", title = "Memory Efficient Hyperelliptic Curve Point Counting", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "203--214", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004034", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004034", abstract = "In recent algorithms that use deformation in order to compute the number of points on varieties over a finite field, certain differential equations of matrices over $p$-adic fields emerge. We present a novel strategy to solve this kind of equations in a memory efficient way. The main application is an algorithm requiring quasi-cubic time and only quadratic memory in the parameter n, that solves the following problem: for E a hyperelliptic curve of genus g over a finite field of extension degree n and small characteristic, compute its zeta function. This improves substantially upon Kedlaya's result which has the same quasi-cubic time asymptotic, but requires also cubic memory size.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Boulet:2011:SMD, author = "Cilanne Boulet and Ka{\u{g}}an Kur{\c{s}}ung{\"o}z", title = "Symmetry of $k$-marked {Durfee} symbols", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "215--230", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003971", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003971", abstract = "Andrews introduced the k-marked Durfee symbols in his work defining a variant of the Atkin--Garvan moments of ranks. He provided and proved many identities and congruences using analytical methods. Here, we give an equivalent description of k-marked Durfee symbols, and using it we give combinatorial proofs to two results of Andrews'.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Brink:2011:RFD, author = "David Brink", title = "{R{\'e}dei} Fields and Dyadic Extensions", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "231--240", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003983", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003983", abstract = "For an arbitrary non-square discriminant D, the {\em R{\'e}dei field\/} \Gamma$_0$ (D) is introduced as an extension of analogous to the genus field and connected with the R{\'e}dei--Reichardt Theorem. It is shown how to compute R{\'e}dei fields, and this is used to find socles of dyadic extensions of K for negative D. Finally, a theorem and two conjectures are presented relating the fields and for an odd prime p.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Li:2011:WWL, author = "Xiaoqing Li", title = "A Weighted {Weyl} Law for the Modular Surface", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "241--248", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003995", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003995", abstract = "In this paper, we will prove a Weyl law for the modular surface weighted by the first Fourier coefficient of the Maass cusp forms. Our error term corresponds to the best known error term in the Weyl law and improves a previous result of Kuznetsov.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baba:2011:OSM, author = "Srinath Baba and H{\aa}kan Granath", title = "Orthogonal Systems of Modular Forms and Supersingular Polynomials", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "249--259", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004009", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004009", abstract = "We extend a construction of Kaneko and Zagier to obtain modular forms which, modulo a prime, vanish at the supersingular points. These modular forms arise simultaneously as solutions of certain second-order differential equations, and as an orthogonal basis for an inner product on the space of modular forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Carter:2011:BGP, author = "Andrea C. Carter", title = "The {Brauer} Group of {Del Pezzo} Surfaces", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "261--287", month = mar, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042111003910", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003910", abstract = "Let S$_1$ be a Del Pezzo surface of degree 1 over a number field k. We establish a criterion for the existence of a nontrivial element of order 5 in the Brauer group of S$_1$ in terms of certain Galois-stable configurations of exceptional divisors on this surface.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Roberts:2011:NPF, author = "David P. Roberts", title = "Nonsolvable Polynomials with Field Discriminant {$ 5^A $}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "289--322", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004113", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004113", abstract = "We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form \pm p$^A$ for p \leq 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL$_2$ (5)$^5$. 10, and field discriminant 5$^{69}$. A closely related polynomial has degree 120, Galois group of the form SL$_2$ (5)$^5$. 20, and field discriminant 5$^{311}$. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 \cdotp 5$^{-1 / 12500}$ \approx 124.984 and the class number of the latter field is divisible by 5$^4$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Murty:2011:TCI, author = "M. Ram Murty and Chester J. Weatherby", title = "On the Transcendence of Certain Infinite Series", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "323--339", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004058", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004058", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Barcau:2011:CCF, author = "Mugurel Barcau and Vicen{\c{t}}iu Pa{\c{s}}ol", title = "$ \bmod p $ congruences for cusp forms of weight four for {$ \Gamma_0 (p N) $}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "341--350", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S179304211100406X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100406X", abstract = "In [1], the authors prove a conjecture of Calegari and Stein regarding $ \bmod p $ congruences between cusp forms of weight four for \Gamma$_0$ (p) and the derivatives of cusp forms of weight two for the same congruence subgroup. In this paper, we investigate whether or not the result remains valid for cusp forms of level N$_p$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Muic:2011:NVC, author = "Goran Mui{\'c}", title = "On the Non-Vanishing of Certain Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "351--370", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004083", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004083", abstract = "Let \Gamma \subset SL$_2$ ({\mathbb{R}}) be a Fuchsian group of the first kind. In this paper, we study the non-vanishing of the spanning set for the space of cuspidal modular forms of weight m \geq 3 constructed in [5, Corollary 2.6.11]. Our approach is based on the generalization of the non-vanishing criterion for L$^1$ Poincar{\'e} series defined for locally compact groups and proved in [6, Theorem 4.1]. We obtain very sharp bounds for the non-vanishing of the spaces of cusp forms for general \Gamma having at least one cusp. We obtain explicit results for congruence subgroups \Gamma (N), \Gamma$_0$ (N), and \Gamma$_1$ (N) (N \geq 1).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bui:2011:CVD, author = "H. M. Bui and Micah B. Milinovich", title = "Central Values of Derivatives of {Dirichlet} {$L$}-Functions", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "371--388", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004125", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004125", abstract = "Let be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method, we show that L$^{(k)}$ (\frac{1}{2}, \chi) \neq 0 for almost all the characters when k and q are large. Here L(s, \chi) is the Dirichlet {$L$}-function associated to the character \chi.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Martin:2011:RTF, author = "Kimball Martin and Mark McKee and Eric Wambach", title = "A Relative Trace Formula for a Compact {Riemann} Surface", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "389--429", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004101", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004101", abstract = "We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic C. This can be expressed as a relation between the period spectrum and the ortholength spectrum of C. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along C as well estimates on the lengths of geodesic segments which start and end orthogonally on C. Variant trace formulas also lead to several simultaneous nonvanishing results for different periods.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bundschuh:2011:AND, author = "Peter Bundschuh and Keijo V{\"a}{\"a}n{\"a}nen", title = "An application of {Nesterenko}'s dimension estimate to $p$-adic $q$-series", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "431--447", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004071", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004071", abstract = "Very recently, Nesterenko proved a $p$-adic analogue of his famous dimension estimate from 1985. The main aim of our present paper is to use this criterion to obtain lower bounds for the dimension of {$ \mathbb {Q}$}-vector spaces spanned by the values at certain rational points of $p$-adic solutions of a class of linear q-difference equations. For the application of Nesterenko's new estimate, we first need a $p$-adic analogue of T{\"o}pfer's results on entire solutions of such functional equations, and secondly, very precise evaluations of certain $p$-adic Schnirelman integrals.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zorn:2011:EDI, author = "Christian Zorn", title = "Explicit doubling integrals for {$ \mathrm {Sp}_2 (F) $} and {$ \widetilde {\mathrm {Sp}_2}(F) $} using ``good test vectors''", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "449--527", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004046", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004046", abstract = "In this paper, we offer some explicit computations of a formulation of the doubling method of Piatetski-Shapiro and Rallis for the groups Sp$_2$ (F) (the rank 2 symplectic group) and its metaplectic cover for F a finite extension of {$ \mathbb {Q}$}$_p$ with p \neq 2. We determine a set of ``good test vectors'' for the irreducible constituents of unramified principal series representations for these groups as well as a set of ``good theta test sections'' in a family of degenerate principal series representations of Sp$_4$ (F) and . Determining ``good test data'' that produces a non-vanishing doubling integral should indicate the existence of a non-vanishing theta lifts for dual pairs of the type (Sp$_2$ (F), O(V)) (respectively) where V is a quadratic space of an even (respectively odd) dimension.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mori:2011:PSE, author = "Andrea Mori", title = "Power Series Expansions of Modular Forms and Their Interpolation Properties", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "2", pages = "529--577", month = mar, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004095", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004095", abstract = "We define a power series expansion of an holomorphic modular form $f$ in the $p$-adic neighborhood of a CM point $x$ of type $K$ for a split good prime $p$. The modularity group can be either a classical conguence group or a group of norm $1$ elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give $p$-adic information on the ring of definition of $f$. By letting the CM point $x$ vary in its Galois orbit, the $r$-th coefficients define a $p$-adic $ K^\times $-modular form in the sense of Hida. By coupling this form with the $p$-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin--Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a $p$-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic $L$-function associated with the base change of $f$ to $K$.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pollack:2011:ESP, author = "Paul Pollack", title = "The Exceptional Set in the Polynomial {Goldbach} Problem", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "579--591", month = may, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042111004423", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004423", abstract = "For each natural number N, let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet {$L$}-functions, that the formula holds ``on average'' in a certain sense. From this they deduced (under ERH) that all but O$_{\epsilon }$ (x$^{1 / 2 + \epsilon }$) of the even natural numbers in [1, x] can be written as a sum of two primes. We generalize their results to the setting of polynomials over a finite field. Owing to Weil's Riemann Hypothesis, our results are unconditional.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bugeaud:2011:MCC, author = "Yann Bugeaud and Alan Haynes and Sanju Velani", title = "Metric Considerations Concerning the Mixed {Littlewood} Conjecture", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "593--609", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004289", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004289", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Draziotis:2011:NIP, author = "Konstantinos A. Draziotis", title = "On the number of integer points on the elliptic curve {$ y^2 = x^3 + A x $}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "611--621", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004149", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004149", abstract = "It is given an upper bound for the number of the integer points of the elliptic curve y$^2$ = x$^3$ + Ax (A \in {\mathbb{Z}}) and a conjecture of Schmidt is proven for this family of elliptic curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Medina:2011:IPL, author = "Luis A. Medina and Victor H. Moll and Eric S. Rowland", title = "Iterated Primitives of Logarithmic Powers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "623--634", month = may, year = "2011", DOI = "https://doi.org/10.1142/S179304211100423X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100423X", abstract = "The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these polynomials is established.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ziegler:2011:AUS, author = "Volker Ziegler", title = "The Additive {$S$}-Unit Structure of Quadratic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "635--644", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004216", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004216", abstract = "We consider a variation of the unit sum number problem for quadratic fields and prove various results.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sun:2011:SNC, author = "Zhi-Wei Sun and Roberto Tauraso", title = "On Some New Congruences for Binomial Coefficients", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "645--662", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004393", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004393", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Astaneh-Asl:2011:IHP, author = "Ali Astaneh-Asl and Hassan Daghigh", title = "Independence of {Heegner} Points for Nonmaximal Orders", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "663--669", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004241", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004241", abstract = "The independence of Heegner points associated to distinct imaginary quadratic fields has been shown by Rosen and Silverman. In this paper we show the independence of Heegner points associated to orders with the same conductor in distinct imaginary quadratic fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gekeler:2011:ZDD, author = "Ernst-Ulrich Gekeler", title = "Zero Distribution and Decay at Infinity of {Drinfeld} Modular Coefficient Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "671--693", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004307", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004307", abstract = "Let \Gamma = GL(2, {$ \mathbb {F} $}$_q$ [T]) be the Drinfeld modular group, which acts on the rigid analytic upper half-plane \Omega. We determine the zeroes of the coefficient modular forms$_a$ \ell$_k$ on the standard fundamental domain for \Gamma on \Omega, along with the dependence of |$_a$ \ell$_k$ (z)| on.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Widmer:2011:NFN, author = "Martin Widmer", title = "On Number Fields with Nontrivial Subfields", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "695--720", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004204", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004204", abstract = "What is the probability for a number field of composite degree d to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if d > 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this paper is an estimate for the number of algebraic numbers of degree d = en and bounded height which generate a field that contains an unspecified subfield of degree e. If n > {maxe$^2$ + e, 10}, we get the correct asymptotics as the height tends to infinity.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Maire:2011:PLE, author = "Christian Maire", title = "Plongements locaux et extensions de corps de nombres. ({French}) [{Local} embeddings and number field extensions]", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "721--738", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004332", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004332", abstract = "Dans ce travail, nous nous int{\'e}ressons au plongement des T-unit{\'e}s d'un corps de nombres K dans une partie de ses compl{\'e}t{\'e}s $p$-adiques construite sur l'ensemble S. Nous montrons que l'injectivit{\'e} de permet d'obtenir des informations sur la structure du groupe de Galois de certaines extensions de K o{\`u} la ramification est li{\'e}e {\`a} S et la d{\'e}composition {\`a} T. Nous {\'e}tudions {\'e}galement le comportement asymptotique du noyau de le long d'une extension $p$-adique analytique sans $p$-torsion. In this article, we are interested in the embedding of the T-units of a number field K in some part of its $p$-adic completions at S. We show that the injectivity of allows us to obtain some information on the structure of the Galois group of some extensions of K where the ramification is attached at S and the decomposition at T. Moreover, we study the asymptotic behavior of the kernel along a $p$-adic analytic extension.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Zywina:2011:RKC, author = "David Zywina", title = "A Refinement of {Koblitz}'s Conjecture", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "739--769", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004411", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004411", abstract = "Let E be an elliptic curve over the rationals. In 1988, Koblitz conjectured an asymptotic for the number of primes $p$ for which the cardinality of the group of {$ \mathbb {F} $}$_p$-points of $E$ is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the $ |E(\mathbb {F}_p)|$ need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often $ |E(\mathbb {F}_p)| / t$ is an integer and prime for a fixed positive integer $t$, and to consider elliptic curves over arbitrary number fields. Several worked out examples are provided to supply numerical evidence for the new conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Odzak:2011:IFG, author = "Almasa Od{\v{z}}ak and Lejla Smajlovi{\'c}", title = "On interpolation functions for generalized {Li} coefficients in the {Selberg} class", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "771--792", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004356", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004356", abstract = "We prove that there exists an entire complex function of order one and finite exponential type that interpolates the Li coefficients \lambda$_F$ (n) attached to a function F in the class that contains both the Selberg class of functions and (unconditionally) the class of all automorphic {$L$}-functions attached to irreducible, cuspidal, unitary representations of GL$_n$ ({$ \mathbb {Q}$}). We also prove that the interpolation function is (essentially) unique, under generalized Riemann hypothesis. Furthermore, we obtain entire functions of order one and finite exponential type that interpolate both archimedean and non-archimedean contribution to \lambda$_F$ (n) and show that those functions can be interpreted as zeta functions built, respectively, over trivial zeros and all zeros of a function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kaavya:2011:CPP, author = "S. J. Kaavya", title = "Crank $0$ Partitions and the Parity of the Partition Function", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "793--801", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004381", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004381", abstract = "A well-known problem regarding the integer partition function p(n) is the {\em parity problem\/}, how often is p(n) even or odd? Motivated by this problem, we obtain the following results: (1) A generating function for the number of crank 0 partitions of n. (2) An involution on the crank 0 partitions whose fixed points are called {\em invariant\/} partitions. We then derive a generating function for the number of invariant partitions. (3) A generating function for the number of self-conjugate rank 0 partitions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gallegos-Ruiz:2011:IPH, author = "Homero R. Gallegos-Ruiz", title = "{$S$}-integral points on hyperelliptic curves", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "803--824", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004435", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004435", abstract = "Let C : Y$^2$ = a$_n$ X$^n$ + \cdots + a$_0$ be a hyperelliptic curve with the a$_i$ rational integers, n \geq 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. We give a completely explicit upper bound for the size of the S-integral points on the model C, provided we know at least one rational point on C and a Mordell--Weil basis for J({$ \mathbb {Q}$}). We use a refinement of the Mordell--Weil sieve which, combined with the upper bound, is capable of determining all the S-integral points. The method is illustrated by determining the S-integral points on the genus 2 hyperelliptic model Y$^2$- Y = X$^5$ X for the set S of the first 22 primes.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bringmann:2011:EFC, author = "Kathrin Bringmann and Olav K. Richter", title = "Exact Formulas for Coefficients of {Jacobi} Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "825--833", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004617", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004617", abstract = "In previous work, we introduced harmonic Maass--Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass--Jacobi--Poincar{\'e} series, as well as Zwegers' real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass--Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass--Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Meemark:2011:DSM, author = "Yotsanan Meemark and Nawaphon Maingam", title = "The Digraph of the Square Mapping on Quotient Rings Over the {Gaussian} Integers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "835--852", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004459", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004459", abstract = "In this work, we investigate the structure of the digraph associated with the square mapping on the ring of Gaussian integers by using the exponent of the unit group modulo \gamma. The formula for the fixed points of is established. Some connections of the lengths of cycles with the exponent of the unit group modulo \gamma are presented. Furthermore, we study the maximum distance from the cycle on each component.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Han:2011:APF, author = "Jeong Soon Han and Hee Sik Kim and J. Neggers", title = "Acknowledgment of priority: {``The Fibonacci-norm of a positive integer: Observations and conjectures''}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "3", pages = "853--854", month = may, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004927", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See \cite{Han:2010:FNP}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004927", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Flicker:2011:CFP, author = "Yuval Z. Flicker", title = "Cusp forms on {$ \mathrm {GSp}(4) $} with {$ \mathrm {SO}(4) $}-periods", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "855--919", month = jun, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042111004186", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004186", abstract = "The Saito--Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the ``relative trace formula''), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F) associated to a quadratic extension E of the global base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture. Technical advances here concern the development of the summation formula and matching of relative orbital integrals.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Schwartz:2011:SAP, author = "Ryan Schwartz and J{\'o}zsef Solymosi and Frank {De Zeeuw}", title = "Simultaneous Arithmetic Progressions on Algebraic Curves", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "921--931", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004198", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004198", abstract = "A {\em simultaneous arithmetic progression\/} (s.a.p.) of length k consists of k points (x$_i$, y$_{\sigma (i)}$), where and are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over {$ \mathbb {Q}$}. We show that 4319 is such a bound for curves over {\mathbb{R}}. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the ``crossing inequality'' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We also extend this method to bound the k for which a real algebraic curve can contain k points from a k $ \times $ k grid. Lastly, these results are extended to complex algebraic curves.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Griffin:2011:DPC, author = "Michael Griffin", title = "Divisibility Properties of Coefficients of Weight $0$ Weakly Holomorphic Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "933--941", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004599", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004599", abstract = "In 1949, Lehner showed that certain coefficients of the modular invariant j(\tau) are divisible by high powers of small primes. Kolberg refined Lehner's results and proved congruences for these coefficients modulo high powers of these primes. We extend Lehner's and Kolberg's work to the elements of a canonical basis for the space of weight 0 weakly holomorphic modular forms.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hohn:2011:TGR, author = "Gerald H{\"o}hn", title = "On a Theorem of {Garza} Regarding Algebraic Numbers with Real Conjugates", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "943--945", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004320", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004320", abstract = "We give a new proof of a theorem on the height of algebraic numbers with real conjugates.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Elsenhans:2011:CSG, author = "Andreas-Stephan Elsenhans and J{\"o}rg Jahnel", title = "Cubic Surfaces with a {Galois} Invariant Pair of {Steiner} Trihedra", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "947--970", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004253", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004253", abstract = "We present a method to construct non-singular cubic surfaces over {$ \mathbb {Q} $} with a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a form generalizing that of Cayley and Salmon. For these, we develop an explicit version of Galois descent.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Saha:2011:PDR, author = "Abhishek Saha", title = "Prime Density Results for {Hecke} Eigenvalues of a {Siegel} Cusp Form", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "971--979", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004642", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004642", abstract = "Let F \in S$_k$ (Sp(2g, {\mathbb{Z}})) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues \mu$_F$ (n). Suppose that the associated automorphic representation \pi$_F$ is locally tempered everywhere. For each c > 0, we consider the set of primes p for which |\mu$_F$ (p)| \geq c and we provide an explicit upper bound on the density of this set. In the case g = 2, we also provide an explicit upper bound on the density of the set of primes p for which \mu$_F$ (p) \geq c.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Miyazaki:2011:TCE, author = "Takafumi Miyazaki", title = "{Terai}'s Conjecture on Exponential {Diophantine} Equations", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "981--999", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004496", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004496", abstract = "Let a, b, c be relatively prime positive integers such that a$^p$ + b$^q$ = c$^r$ with fixed integers p, q, r \geq 2. Terai conjectured that the equation a$^x$ + b$^y$ = c$^z$ has no positive integral solutions other than (x, y, z) = (p, q, r) except for specific cases. Most known results on this conjecture concern the case where p = q = 2 and either r = 2 or odd r \geq 3. In this paper, we consider the case where p = q = 2 and r > 2 is even, and partially verify Terai's conjecture.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Billerey:2011:CIP, author = "Nicolas Billerey", title = "Crit{\`e}res d'irr{\'e}ductibilit{\'e} pour les repr{\'e}sentations des courbes elliptiques. ({French}) [{Irreducibility} criteria for the representations of elliptic curves]", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "1001--1032", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004538", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004538", abstract = "Soit E une courbe elliptique d{\'e}finie sur un corps de nombres K. On dit qu'un nombre premier p est r{\'e}ductible pour le couple (E, K) si E admet une $p$-isog{\'e}nie d{\'e}finie sur K. L'ensemble de tous ces nombres premiers est fini si et seulement si E n'a pas de multiplication complexe d{\'e}finie sur K. Dans cet article, on montre que l'ensemble des nombres premiers r{\'e}ductibles pour le couple (E, K) est contenu dans l'ensemble des diviseurs premiers d'une liste explicite d'entiers (d{\'e}pendant de E et de K) dont une infinit{\'e} d'entre eux est non nulle. Cela fournit un algorithme efficace de calcul dans le cas fini. D'autres crit{\`e}res moins g{\'e}n{\'e}raux, mais n{\'e}anmoins utiles sont donn{\'e}s ainsi que de nombreux exemples num{\'e}riques. Let E be an elliptic curve defined over a number field K. We say that a prime number p is reducible for (E, K) if E admits a $p$-isogeny defined over K. The so-called reducible set of all such prime numbers is finite if and only if E does not have complex multiplication over K. In this paper, we prove that the reducible set is included in the set of prime divisors of an explicit list of integers (depending on E and K), infinitely many of them being non-zero. It provides an efficient algorithm for computing it in the finite case. Other less general but rather useful criteria are given, as well as many numerical examples.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Hassen:2011:ZFR, author = "Abdul Hassen and Hieu D. Nguyen", title = "A Zero-Free Region for Hypergeometric Zeta Functions", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "1033--1043", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004678", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004678", abstract = "This paper investigates the location of ``trivial'' zeros of some hypergeometric zeta functions. Analogous to Riemann's zeta function, we demonstrate that they possess a zero-free region on a left-half complex plane, except for infinitely many zeros regularly spaced on the negative real axis.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ghosh:2011:KGT, author = "Anish Ghosh", title = "A {Khintchine--Groshev} theorem for affine hyperplanes", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "1045--1064", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004228", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004228", abstract = "We prove the divergence case of the Khintchine--Groshev theorem for a large class of affine hyperplanes, completing the convergence case proved in [11] and answering in part a question of Beresnevich {\em et al.\/} ([4]). We use the mechanism of regular systems developed in [4] and estimates from [11].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gun:2011:AIV, author = "Sanoli Gun and M. Ram Murty and Purusottam Rath", title = "Algebraic Independence of Values of Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "1065--1074", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004769", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004769", abstract = "We investigate values of modular forms with algebraic Fourier coefficients at algebraic arguments. As a consequence, we conclude about the nature of zeros of such modular forms. In particular, the singular values of modular forms (that is, values at CM points) are related to the recent work of Nesterenko. As an application, we deduce the transcendence of critical values of certain Hecke $L$-series. We also discuss how these investigations generalize to the case of quasi-modular forms with algebraic Fourier coefficients.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Mera:2011:ZFR, author = "Mitsugu Mera", title = "Zero-free regions of a $q$-analogue of the complete {Riemann} zeta function", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "1075--1092", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004344", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004344", abstract = "A q-analogue of the complete Riemann zeta function presented in this paper is defined by the q-Mellin transform of the Jacobi theta function. We study zero-free regions of the q-zeta function. As a by-product, we show that the Riemann zeta function does not vanish in a sub-region of the critical strip.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Cao:2011:SDR, author = "Wei Cao", title = "A Special Degree Reduction of Polynomials Over Finite Fields with Applications", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "1093--1102", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004277", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004277", abstract = "Let f be a polynomial in n variables over the finite field {$ \mathbb {F} $}$_q$ and N$_q$ (f) denote the number of {$ \mathbb {F} $}$_q$-rational points on the affine hypersurface f = 0 in {$ \mathbb {A} $}$^n$ ({$ \mathbb {F} $}$_q$). A \phi -reduction of f is defined to be a transformation \sigma : {$ \mathbb {F} $}$_q$ [x$_1$, \ldots, x$_n$ ] \rightarrow {$ \mathbb {F} $}$_q$ [x$_1$, \ldots, x$_n$ ] such that N$_q$ (f) = N$_q$ (\sigma(f)) and deg f \geq deg \sigma(f). In this paper, we investigate \phi -reduction by using the degree matrix which is formed by the exponents of the variables of f. With \phi -reduction, we may improve various estimates on N$_q$ (f) and utilize the known results for polynomials with low degree. Furthermore, it can be used to find the explicit formula for N$_q$ (f).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Raji:2011:ECT, author = "Wissam Raji", title = "{Eichler} Cohomology Theorem for Generalized Modular Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "4", pages = "1103--1113", month = jun, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004514", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:25 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004514", abstract = "We show starting with relations between Fourier coefficients of weakly parabolic generalized modular forms of negative weight that we can construct automorphic integrals for large integer weights. We finally prove an Eichler isomorphism theorem for weakly parabolic generalized modular forms using the classical approach as in [3].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Freiman:2011:SSC, author = "Gregory A. Freiman and Yonutz V. Stanchescu", title = "Sets with Several Centers of Symmetry", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1115--1135", month = aug, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042111004174", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004174", abstract = "Let A be a finite subset of the group {\mathbb{Z}}$^2$. Let C = {c$_0$, c$_1$, \ldots, c$_{s - 1}$ } be a finite set of s distinct points in the plane. For every 0 \leq i \leq s -1, we define D$_i$ = {a - a\prime : a \in A, a\prime \in A, a + a\prime = 2c$_i$ } and R$_s$ (A) = |D$_0$ \cup D$_1$ \cup \ldots \cup D$_{s - 1}$ |. In [1, 2], we found the maximal value of R$_s$ (A) in cases s = 1, s = 2 and s = 3 and studied the structure of A assuming that R$_3$ (A) is equal or close to its maximal value. In this paper, we examine the case of s = 4 centers of symmetry and we find the {\em maximal value\/} of R$_4$ (A). Moreover, in cases when the maximal value is attained, we will describe the {\em structure of extremal sets}.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Tu:2011:ONA, author = "Fang-Ting Tu", title = "On Orders of {$ M(2, K) $} Over a Non--{Archimedean} Local Field", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1137--1149", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004654", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004654", abstract = "Let K be a non-Archimedean local field. In this paper, we first show that if an order in M(2, K) is the intersection of (finitely many) maximal orders in M(2, K), then it is the intersection of at most three maximal orders. Using this result, we obtain a complete classification of orders in M(2, K) that are intersections of maximal orders.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dixit:2011:ATF, author = "Atul Dixit", title = "Analogues of a Transformation Formula of {Ramanujan}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1151--1172", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S179304211100454X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100454X", abstract = "We derive two new analogues of a transformation formula of Ramanujan involving the Gamma function and Riemann zeta function present in his \booktitle{Lost Notebook}. Both involve infinite series consisting of Hurwitz zeta functions and yield modular-type relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by A. P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ayad:2011:CDV, author = "Mohamed Ayad and Omar Kihel", title = "Common Divisors of Values of Polynomials and Common Factors of Indices in a Number Field", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1173--1194", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004526", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004526", abstract = "Let K be a number field of degree n over {$ \mathbb {Q} $}, {\^A} be the set of integers of K that are primitive over {$ \mathbb {Q} $} and let I(K) be its index. The prime factors of I(K) are called common factors of indices or inessential discriminant divisors. We show that these primes divide another index i(K) previously defined by Gunji and McQuillan as i(K) = lcm$_{\theta \in {\^ A}}$ i(\theta), where i(\theta) = gcd$_{x \in {\mathbb {Z}}}$ F$_{\theta }$ (x) and F$_{\theta }$ (x) is the characteristic polynomial of \theta over {$ \mathbb {Q}$}. It is shown that there exists \theta \in {\^A} such that i(K) = i(\theta) and an algorithm is given for the computation of such an integer. For any prime p|i(K), an integer \rho$_K$ (p) defined as the number of such that p|i(\theta) is investigated. It is shown that this integer determines in some cases the splitting type of p in K. Some open questions related to I(K), i(K) and \rho$_K$ (p) are stated.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Philippon:2011:AFC, author = "Patrice Philippon", title = "Approximations fonctionnelles des courbes des espaces projectifs. ({French}) [{Functional} approximations of the curves of projective spaces]", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1195--1215", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004502", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004502", abstract = "Algebraic approximation to points in projective spaces offers a new and more flexible approach to algebraic independence theory. When working over the field of algebraic numbers, it leads to open conjectures in higher dimension extending known results in Diophantine approximation. We show here that over the algebraic closure of a function field in one variable, the analog of these conjectures is true. We also derive transfer lemmas which have applications in the study of multiplicity estimates, for example.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Dubickas:2011:RPD, author = "Art{\=u}ras Dubickas", title = "Roots of Polynomials with Dominant Term", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1217--1228", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004575", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004575", abstract = "We characterize all algebraic numbers which are roots of integer polynomials with a coefficient whose modulus is greater than or equal to the sum of moduli of all the remaining coefficients. It turns out that these numbers are zero, roots of unity and those algebraic numbers \beta whose conjugates over {$ \mathbb {Q} $} (including \beta itself) do not lie on the circle |z| = 1. We also describe all algebraic integers with norm B which are roots of an integer polynomial with constant coefficient B and the sum of moduli of all other coefficients at most |B|. In contrast to the above, the set of such algebraic integers is ``quite small''. These results are motivated by a recent paper of Frougny and Steiner on the so-called minimal weight \beta -expansions and are also related to some work on canonical number systems and tilings.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Boylan:2011:VFC, author = "Matthew Boylan and Sharon Anne Garthwaite and John Webb", title = "On the Vanishing of {Fourier} Coefficients of Certain Genus Zero Newforms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1229--1245", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004290", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004290", abstract = "Given a classical modular form f(z), a basic question is whether any of its Fourier coefficients vanish. This question remains open for certain modular forms. For example, let \Delta (z) = \Sigma \Gamma (n)q$^n$ \in S$_{12}$ (\Gamma$_0$ (1)). A well-known conjecture of Lehmer asserts that \tau (n) \neq 0 for all n. In recent work, Ono constructed a family of polynomials A$_n$ (x) \in {$ \mathbb {Q}$}[x] with the property that \tau (n) vanishes if and only if A$_n$ (0) and A$_n$ (1728) do. In this paper, we establish a similar criterion for the vanishing of coefficients of certain newforms on genus zero groups of prime level.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ghitza:2011:DHE, author = "Alexandru Ghitza", title = "Distinguishing {Hecke} Eigenforms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1247--1253", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004708", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004708", abstract = "We revisit a theorem of Ram Murty about the number of initial Fourier coefficients that two cuspidal eigenforms of different weights can have in common. We prove an explicit upper bound on this number, and give better conditional and unconditional asymptotic upper bounds. Finally, we describe a numerical experiment testing the sharpness of the upper bound in the case of forms of level one.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Williams:2011:SFO, author = "H. C. Williams and R. K. Guy", title = "Some Fourth-Order Linear Divisibility Sequences", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1255--1277", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004587", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004587", abstract = "We extend the Lucas--Lehmer theory for second-order divisibility sequences to a large class of fourth-order sequences, with appropriate laws of apparition and of repetition. Examples are provided by the numbers of perfect matchings, or of spanning trees, in families of graphs, and by the numbers of points on elliptic curves over finite fields. Whether there are fourth-order divisibility sequences not covered by our theory is an open question.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Liu:2011:GUN, author = "Huaning Liu", title = "{Gowers} Uniformity Norm and Pseudorandom Measures of the Pseudorandom Binary Sequences", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1279--1302", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004137", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004137", abstract = "Recently there has been much progress in the study of arithmetic progressions. An important tool in these developments is the Gowers uniformity norm. In this paper we study the Gowers norm for pseudorandom binary sequences, and establish some connections between these two subjects. Some examples are given to show that the ``good'' pseudorandom sequences have small Gowers norm. Furthermore, we introduce two large families of pseudorandom binary sequences constructed by the multiplicative inverse and additive character, and study the pseudorandom measures and the Gowers norm of these sequences by using the estimates of exponential sums and properties of the Vandermonde determinant. Our constructions are superior to the previous ones from some points of view.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Dahmen:2011:RMA, author = "Sander R. Dahmen", title = "A refined modular approach to the {Diophantine} equation $ x^2 + y^{2 n} = z^3 $", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1303--1316", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004472", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004472", abstract = "Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x$^2$ + y$^{2n}$ = z$^3$ in nonzero coprime integer unknowns x,y,z. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for n \in {5,31} there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for n \leq 10$^7$. Finally, we show that there are also no solutions for n \equiv -1 (mod 6).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Liu:2011:DTS, author = "Zhixin Liu and Guangshi L{\"u}", title = "Density of Two Squares of Primes and Powers of $2$", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1317--1329", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004605", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004605", abstract = "As the generalization of the problem of Romanoff, we establish that a positive proportion of integers can be written as the sum of two squares of primes and two powers of 2. We also prove that every large even integer can be written as the sum of two primes and 12 powers of 2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Baruah:2011:SCD, author = "Nayandeep Deka Baruah and Kanan Kumari Ojah", title = "Some Congruences Deducible from {Ramanujan}'s Cubic Continued Fraction", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1331--1343", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004745", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004745", abstract = "We present some interesting Ramanujan-type congruences for some partition functions arising from Ramanujan's cubic continued fraction. One of our results states that if p$_3$ (n) is defined by, then p$_3$ (9n + 8) \equiv 0 (mod 3$^4$).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Harada:2011:CSU, author = "Masaaki Harada", title = "Construction of Some Unimodular Lattices with Long Shadows", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1345--1358", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004794", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004794", abstract = "In this paper, we construct odd unimodular lattices in dimensions n = 36,37 having minimum norm 3 and 4s = n - 16, where s is the minimum norm of the shadow. We also construct odd unimodular lattices in dimensions n = 41,43,44 having minimum norm 4 and 4s = n - 24.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Wang:2011:API, author = "Xinna Wang and Yingchun Cai", title = "An Additive Problem Involving {Piatetski--Shapiro} Primes", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1359--1378", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004630", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004630", abstract = "Let P$_r$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that there exist infinitely many primes of the form p = [n$^c$ ] such that p + 2 = P$_r$, where r is the least positive integer satisfying certain inequalities. In particular for we have r = 5. This result constitutes an improvement upon that of T. P. Peneva.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Park:2011:ALG, author = "Jeehoon Park", title = "Another look at {Gross--Stark} units over the number field {$ \mathbb {Q} $}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1379--1393", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004150", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004150", abstract = "We provide another description of the Gross--Stark units over the rational field {$ \mathbb {Q} $} (studied in [B. Gross, $p$-adic $L$-series at s = 0, {\em J. Fac. Sci. Univ. Tokyo\/} 28(3) (1981) 979--994]) which is essentially a Gauss sum, using a $p$-adic multiplicative integral of the {\em $p$-adic Kubota--Leopoldt distribution\/}, and give a simplified proof of the Ferrero--Greenberg theorem (see [B. Ferrero and R. Greenberg, On the behavior of $p$-adic {$L$}-functions at s = 0, {\em Invent. Math.\/} 50(1) (1978/79) 91--102]) for $p$-adic Hurwitz zeta functions. This is a precise analog for {$ \mathbb {Q}$} of Darmon--Dasgupta's work on {\em elliptic units for real quadratic fields\/} (see [H. Darmon and S. Dasgupta, Elliptic units for real quadratic fields, {\em Ann. of Math. (2)\/} 163(1) (2006) 301--346]).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ryan:2011:BTC, author = "Nathan C. Ryan and Gonzalo Tornar{\'i}a", title = "A {B{\"o}cherer}-type conjecture for paramodular forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "5", pages = "1395--1411", month = aug, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004629", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004629", abstract = "In the 1980s B{\"o}cherer formulated a conjecture relating the central value of the quadratic twists of the spinor {$L$}-function attached to a Siegel modular form F to the coefficients of F. He proved the conjecture when F is a Saito--Kurokawa lift. Later Kohnen and Ku{\ss} gave numerical evidence for the conjecture in the case when F is a rational eigenform that is not a Saito--Kurokawa lift. In this paper we develop a conjecture relating the central value of the quadratic twists of the spinor {$L$}-function attached to a paramodular form and the coefficients of the form. We prove the conjecture in the case when the form is a Gritsenko lift and provide numerical evidence when it is not a lift.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Goldston:2011:JCG, author = "D. A. Goldston and A. H. Ledoan", title = "Jumping Champions and Gaps Between Consecutive Primes", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1413--1421", month = sep, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S179304211100471X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100471X", abstract = "The most common difference that occurs among the consecutive primes less than or equal to x is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given x. In 1999 Odlyzko, Rubinstein and Wolf provided heuristic and empirical evidence in support of the conjecture that the numbers greater than 1 that are jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,\ldots. As a step toward proving this conjecture they introduced a second weaker conjecture that any fixed prime p divides all sufficiently large jumping champions. In this paper we extend a method of Erd{\H{o}}s and Straus from 1980 to prove that the second conjecture follows directly from the prime pair conjecture of Hardy and Littlewood.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Djankovic:2011:NFA, author = "Goran Djankovi{\'c}", title = "Nonvanishing of the family of {$ \Gamma_1 (q) $}-automorphic {$L$}-functions at the central point", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1423--1439", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004800", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004800", abstract = "We investigate proportion of nonvanishing central values L(f, 1/2) of {$L$}-functions associated to the basis of holomorphic modular forms of fixed weight k with respect to \Gamma$_1$ (q), in the limit when q \rightarrow \infty along the primes. Motivation is a contrast between \Gamma$_0$ (q) and \Gamma$_1$ (q) families in the sense of underlying harmonic analysis.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Komori:2011:FED, author = "Yasushi Komori and Kohji Matsumoto and Hirofumi Tsumura", title = "Functional Equations for Double {$L$}-Functions and Values at Non-Positive Integers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1441--1461", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004551", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004551", abstract = "We consider double {$L$}-functions with periodic coefficients and complex parameters. We prove functional equations for them, which is of traditional symmetric form on certain hyperplanes. These are character analogs of our previous result on double zeta-functions. We further evaluate double {$L$}-functions at non-positive integers and construct certain $p$-adic double {$L$}-functions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Gao:2011:QAN, author = "Weidong Gao and Alfred Geroldinger and Qinghong Wang", title = "A Quantitative Aspect of Non-Unique Factorizations: the {Narkiewicz} Constants", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1463--1502", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004721", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004721", abstract = "Let K be an algebraic number field with non-trivial class group G and let be its ring of integers. For k \in \mathbb{N} and some real x \geq 1, let F$_k$ (x) denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that F$_k$ (x) behaves, for x \rightarrow \infty, asymptotically like x(log x)$^{-1 + 1 / |G|}$ (log log x)$^{N k (G)}$. We study N$_k$ (G) with new methods from Combinatorial Number Theory.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pitoun:2011:CCM, author = "Fr{\'e}d{\'e}ric Pitoun", title = "Conoyaux de capitulation et modules d'{Iwasawa}. ({French}) [{Capitulation} co-kernels and {Iwasawa} modules]", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1503--1517", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004861", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004861", abstract = "Soit F un corps de nombres totalement r{\'e}el et p un premier impair, on note $ K_0 $ = F(\zeta$_p$). Pour n \in \mathbb{N}, $ K_n$ d{\'e}signe le n-i{\`e}me {\'e}tage de la {\mathbb{Z}}$_p$ extension cyclotomique $ K_{\infty }$ /K$_0$, A$_n$ est la $p$-partie du groupe des classes de $ K_n$, et N$_{\infty }$ est l'extension de $ K_{\infty }$ obtenue en extrayant des racines $p$-primaires d'unit{\'e}s. Le but de cet article est de montrer que le dual de Pontryagin de la partie plus des conoyaux de capitulation, sur laquelle l'action de \Gamma a {\'e}t{\'e} tordue une fois par le caract{\`e}re cyclotomique et la partie moins de la {\mathbb{Z}}$_p$ torsion du groupe de Galois Gal(N$_{\infty }$ \cap L$_{\infty }$ /K$_{\infty }$) sont isomorphes. Let F be a totally real number field and p an odd prime, we note $ K_0$ = F(\zeta$_p$). For an integer n, $ K_n$ is the nth floor of the {\mathbb{Z}}$_p$-cyclotomic extension $ K_{\infty }$ /K$_0$, A$_n$ is the $p$-part of the class group of $ K_n$, and N$_{\infty }$ is the extension of $ K_{\infty }$ generated by $p$-primary roots of units. In this article, we prove that the plus part of the capitulation's cokernel, on which \Gamma -action was twisted on time by the cyclotomic character, and the minus part of the {\mathbb{Z}}$_p$-torsion of the Galois group Gal(N$_{\infty }$ \cap L$_{\infty }$ /K$_{\infty }$) is isomorphic.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", language = "French", } @Article{Blache:2011:NPC, author = "R{\'e}gis Blache", title = "{Newton} Polygons for Character Sums and {Poincar{\'e}} Series", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1519--1542", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004368", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004368", abstract = "In this paper, we precise the asymptotic behavior of Newton polygons of {$L$}-functions associated to character sums, coming from certain n variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum \Delta = \Delta$_1$ \oplus \Delta$_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behavior of Newton polygons for multivariable polynomials when the generic Newton polygon differs from the combinatorial (Hodge) polygon associated to the polyhedron.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Girstmair:2011:CFJ, author = "Kurt Girstmair", title = "Continued Fractions and {Jacobi} Symbols", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1543--1555", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004848", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004848", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ostafe:2011:MCS, author = "Alina Ostafe and Igor E. Shparlinski and Arne Winterhof", title = "Multiplicative Character Sums of a Class of Nonlinear Recurrence Vector Sequences", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1557--1571", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004484", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004484", abstract = "We estimate multiplicative character sums along the orbits of a class of nonlinear recurrence vector sequences. In the one-dimensional case, only much weaker estimates are known and our results have no one-dimensional analogs.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Papikian:2011:GAG, author = "Mihran Papikian", title = "On Generators of Arithmetic Groups Over Function Fields", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1573--1587", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004265", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004265", abstract = "Let F = {$ \mathbb {F} $}$_q$ (T) be the field of rational functions with {$ \mathbb {F} $}$_q$-coefficients, and A = {$ \mathbb {F} $}$_q$ [T] be the subring of polynomials. Let D be a division quaternion algebra over F which is split at 1/T. For certain A-orders in D we find explicit finite sets generating their groups of units.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Xia:2011:DRG, author = "Ernest X. W. Xia and X. M. Yao", title = "The $8$-dissection of the {Ramanujan--G{\"o}llnitz--Gordon} continued fraction by an iterative method", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1589--1593", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004824", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004824", abstract = "In this paper, we present an iterative method to derive the 8-dissections of the Ramanujan--G{\"o}llnitz--Gordon continued fraction and its reciprocal which were first discovered by Hirschhorn.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chen:2011:CP, author = "Yong-Gao Chen and Jing-Rui Lou", title = "The congruent properties for $ r_s(n) $", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1595--1602", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004885", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", note = "See erratum \cite{Chen:2013:ECP}.", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004885", abstract = "Let r$_s$ (n) denote the number of ways to write an integer n as the sum of s squares of integers. In this paper, the congruent properties for r$_s$ (n) are studied. We give the elementary combinatorial proofs of all related results due to Wagstaff and Chen, and obtain some new results.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Oh:2011:RAP, author = "Byeong-Kweon Oh", title = "Representations of Arithmetic Progressions by Positive Definite Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1603--1614", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004915", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004915", abstract = "For a positive integer d and a non-negative integer a, let S$_{d, a}$ be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be S$_{d, a}$- {\em universal\/} if it represents all integers in the set S$_{d, a}$, and is said to be S$_{d, a}$- {\em regular\/} if it represents all integers in the non-empty set S$_{d, a}$ \cap Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) \in {$ \mathbb {Q}$}[x,y] (R(x,y) \in {$ \mathbb {Q}$}[x,y]) such that the discriminant df for any S$_{d, a}$ universal (S$_{d, a}$-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Xiong:2011:TII, author = "Xinhua Xiong", title = "Two Identities Involving the Cubic Partition Function", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1615--1626", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004757", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004757", abstract = "We give a very elementary proof of an identity involving the cubic partition function and we also give an elementary proof of a new identity for the cubic partition function which is analogs to Zuckerman's identity for the ordinary partition function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Fukuda:2011:WCN, author = "Takashi Fukuda and Keiichi Komatsu", title = "{Weber}'s class number problem in the cyclotomic {$ \mathbb {Z}_2 $}-extension of {$ \mathbb {Q} $}, {III}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1627--1635", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004782", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004782", abstract = "Let h$_n$ denote the class number of which is a cyclic extension of degree 2$^n$ over the rational number field {$ \mathbb {Q}$}. There are no known examples of h$_n$ > 1. We prove that a prime number \ell does not divide h$_n$ for all n \geq 1 if \ell is less than 10$^9$ or \ell satisfies a congruence relation \ell \nequiv \pm 1 (mod 32).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Sakai:2011:AOP, author = "Yuichi Sakai", title = "The {Atkin} Orthogonal Polynomials for the Low-Level {Fricke} Groups and Their Application", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1637--1661", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004460", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004460", abstract = "Kaneko and Zagier proved the relation between the Atkin orthogonal polynomials and supersingular j-polynomials for PSL$_2$ ({\mathbb{Z}}). Tsutsumi also proved its relation for the Hecke subgroups of level less than or equal to 4. In this paper, we define the Atkin inner product for the Fricke groups of level less than or equal to 3 and construct the Atkin orthogonal polynomials. Then, we give the relation between supersingular -polynomials defined by Koike and its polynomials. We also give extremal quasimodular forms of depth 1 by using its orthogonal polynomials.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Turkelli:2011:CMC, author = "Seyfi T{\"u}rkelli", title = "Counting multisections in conic bundles over a curve defined over {$ \mathbb {F}_q $}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1663--1680", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S179304211100485X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100485X", abstract = "For a given conic bundle X over a curve C defined over {$ \mathbb {F} $}$_q$, we count irreducible branch covers of C in X of degree d and height e \gg 1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field {$ \mathbb {F} $}$_q$ (C).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hong:2011:ABS, author = "Shaofang Hong and Raphael Loewy", title = "Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions $ (\bmod r) $", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "6", pages = "1681--1704", month = sep, year = "2011", DOI = "https://doi.org/10.1142/S179304211100437X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100437X", abstract = "In this paper, we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod r). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Knopfmacher:2011:PPN, author = "Arnold Knopfmacher and Florian Luca", title = "On Prime-Perfect Numbers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1705--1716", month = nov, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042111004447", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004447", abstract = "We prove that the Diophantine equation has only finitely many positive integer solutions k, p$_1$, \ldots, p$_k$, r$_1$, \ldots, r$_k$, where p$_1$, \ldots, p$_k$ are distinct primes. If a positive integer n has prime factorization, then represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ballot:2011:MD, author = "Christian Ballot and Mireille Car", title = "On {Murata} Densities", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1717--1736", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S179304211100440X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100440X", abstract = "In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, {\em Proc. Japan Acad. Ser. A Math. Sci.\/} 56(7) (1980) 351--353; On some fundamental relations among certain asymptotic densities, {\em Math. Rep. Toyama Univ.\/} 4(2) (1981) 47--61], Murata classified certain prime density functions in the case of the arithmetical semigroup of natural numbers. Here, it is shown that the same density functions will obey a very similar classification in any arithmetical semigroup whose sequence of norms satisfies certain general growth conditions. In particular, this classification holds for the set of monic polynomials in one indeterminate over a finite field, or for the set of ideals of the ring of S-integers of a global function field (S finite).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Hare:2011:SD, author = "Kevin G. Hare and Shanta Laishram and Thomas Stoll", title = "The sum of digits of $n$ and $ n^2$", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1737--1752", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004319", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004319", abstract = "Let s$_q$ (n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s$_2$ (n) = s$_2$ (n$^2$). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case p(n) = n$^2$, looking at the subsets of n where s$_q$ (n) = s$_q$ (n$^2$) = k for fixed k.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bazzanella:2011:TCR, author = "Danilo Bazzanella", title = "Two Conditional Results About Primes in Short Intervals", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1753--1759", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004563", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004563", abstract = "In 1937, Ingham proved that \psi (x + x$^{\theta }$) - \psi (x) \sim x$^{\theta }$ for x \rightarrow \infty, under the assumption of the Lindel{\"o}f hypothesis for \theta > 1/2. In this paper we examine how the above asymptotic formula holds by assuming in turn two different heuristic hypotheses. It must be stressed that both the hypotheses are implied by the Lindel{\"o}f hypothesis.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Ribenboim:2011:MPT, author = "Paulo Ribenboim", title = "Multiple patterns of $k$-tuples of integers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1761--1779", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004733", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004733", abstract = "The first proposition and its corollary are a transfiguration of Dirichlet's pigeon-hole principle. They are applied to show that a wide variety of sequences display arbitrarily large patterns of sums, differences, higher differences, etc. Among these, we include sequences of primes in arithmetic progressions, of powerful integers, sequences of integers with radical index having a prescribed lower bound, and many others. We also deal with patterns in iterated sequences of primes, patterns of gaps between primes, patterns of values of Euler's \phi -function, or their gaps, as well as patterns related to the sequence of Carmichael numbers.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Faber:2011:NRI, author = "Xander Faber and Benjamin Hutz and Michael Stoll", title = "On the Number of Rational Iterated Preimages of the Origin Under Quadratic Dynamical Systems", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1781--1806", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004162", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004162", abstract = "For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform bounds on pre-images under quadratic dynamical systems,'' by two of the present authors and five others, it was shown that the number of rational iterated preimages of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the $L$-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated preimages is six. We also provide further insight into the geometry of the ``preimage curves.''", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chambert-Loir:2011:TJS, author = "Antoine Chambert-Loir", title = "The Theorem of {Jentzsch--Szeg{\H{o}}} on an Analytic Curve: Application to the Irreducibility of Truncations of Power Series", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1807--1823", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004691", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004691", abstract = "A theorem of Jentzsch--Szeg{\H{o}} describes the limit measure of a sequence of discrete measures associated to zeroes of a sequence of polynomials in one variable. Following the presentation by Andrievskii and Blatt in [ {\em Discrepancy of Signed Measures and Polynomial Approximation\/}, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] we extend this theorem to compact Riemann surfaces and to analytic curves in the sense of Berkovich over ultrametric fields, using classical potential theory in the former case, and Baker/Rumely, Thuillier's potential theory on analytic curves in the latter case. We then apply this equidistribution theorem to the question of irreducibility of truncations of power series with coefficients in ultrametric fields. {\em R{\'e}sum{\'e} fran{\c{c}}ais\/}: Le th{\'e}or{\`e}me de Jentzsch--Szeg{\H{o}} d{\'e}crit la mesure limite d'une suite de mesures discr{\`e}tes associ{\'e}e aux z{\'e}ros d'une suite convenable de polyn{\^o}mes en une variable. Suivant la pr{\'e}sentation que font Andrievskii et Blatt dans [ {\em Discrepancy of Signed Measures and Polynomial Approximation\/}, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] on {\'e}tend ici ce r{\'e}sultat aux surfaces de Riemann compactes, puis aux courbes analytiques sur un corps ultram{\'e}trique. On donne pour finir quelques corollaires du cas particulier de la droite projective sur un corps ultram{\'e}trique {\`a} l'irr{\'e}ductibilit{\'e} des polyn{\^o}mes-sections d'une s{\'e}rie enti{\`e}re en une variable.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Berger:2011:AFD, author = "Laurent Berger", title = "A $p$-adic family of dihedral {$ (\phi, \Gamma)$}-modules", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1825--1834", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004770", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004770", abstract = "The goal of this paper is to construct explicitly a $p$-adic family of representations (which are dihedral representations), to construct their attached (\phi, \Gamma)-modules by writing down explicit matrices for \phi and for the action of \Gamma, and finally to determine which of these are trianguline.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Zumalacarregui:2011:CPM, author = "Ana Zumalac{\'a}rregui", title = "Concentration of Points on Modular Quadratic Forms", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1835--1839", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004897", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004897", abstract = "Let Q(x, y) be a quadratic form with discriminant D \neq 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) \equiv \lambda (mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - \lambda is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is M$^{o(1)}$ when M \ll p$^{1 / 4}$. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy \equiv \lambda (mod p).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Petersen:2011:EAN, author = "Kathleen L. Petersen and Christopher D. Sinclair", title = "Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1841--1861", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004666", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004666", abstract = "Given a fixed quadratic extension K of {$ \mathbb {Q} $}, we consider the distribution of elements in K of norm one (denoted ). When K is an imaginary quadratic extension, is naturally embedded in the unit circle in {\mathbb{C}} and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in can be written as for some, which yields another ordering of given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that is equidistributed with respect to norm, under the map \beta \mapsto log|\beta |(mod log|\epsilon$^2$ |) where \epsilon is a fundamental unit of.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Reznick:2011:STC, author = "Bruce Reznick and Jeremy Rouse", title = "On the Sums of Two Cubes", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1863--1882", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004903", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004903", abstract = "We solve the equation f(x, y)$^3$ + g(x, y)$^3$ = x$^3$ + y$^3$ for homogeneous f, g \in {\mathbb{C}}(x, y), completing an investigation begun by Vi{\`e}te in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring isomorphic to {\mathbb{Z}}[e$^{2 \pi i / 3}$ ].", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bouganis:2011:NAC, author = "Thanasis Bouganis", title = "Non--{Abelian} Congruences Between Special Values of {$L$}-Functions of Elliptic Curves: the {CM} Case", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1883--1934", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S179304211100468X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100468X", abstract = "In this work we prove congruences between special values of {$L$}-functions of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analog for elliptic curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive. The proof is based on the fact that the critical values of elliptic curves with CM, or what amounts to the same, the critical values of Gr{\"o}ssencharacters, can be expressed as values of Hilbert--Eisenstein series at CM points. We believe that our strategy can be generalized to provide congruences for a large class of $L$-values.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Li:2011:DCU, author = "Yan Li and Lianrong Ma", title = "Double Coverings and Unit Square Problems for Cyclotomic Fields", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1935--1944", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004836", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004836", abstract = "In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for, where K = {$ \mathbb {Q} $}(\zeta$_n$), G = Gal(K/{$ \mathbb {Q}$}), {$ \mathbb {F} $}$_2$ = {\mathbb{Z}}/2{\mathbb{Z}} and U$_K$ is the unit group of K. We explicitly determine all the cyclotomic fields K = {$ \mathbb {Q}$}(\zeta$_n$) such that . Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, {\em J. Number Theory\/} 128 (2008) 2687--2694]. In particular, we prove that the unit square problem does not hold for {$ \mathbb {Q}$}(\zeta$_n$) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields {$ \mathbb {Q}$}(\zeta$_n$)$_v$ with v | p but not a square in {$ \mathbb {Q}$}(\zeta$_n$), if n has more than three distinct prime factors.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Huber:2011:DEC, author = "Tim Huber", title = "Differential Equations for Cubic Theta Functions", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1945--1957", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004873", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004873", abstract = "We show that the cubic theta functions satisfy two distinct coupled systems of nonlinear differential equations. The resulting relations are analogous to Ramanujan's differential equations for Eisenstein series on the full modular group. We deduce the cubic analogs presented here from trigonometric series identities arising in Ramanujan's original paper on Eisenstein series. Several consequences of these differential equations are established, including a short proof of a famous cubic theta function identity derived by J. M. Borwein and P. B. Borwein.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Guo:2011:FSA, author = "Victor J. W. Guo and Jiang Zeng", title = "Factors of Sums and Alternating Sums Involving Binomial Coefficients and Powers of Integers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "7", pages = "1959--1976", month = nov, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004812", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:26 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004812", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Merila:2011:NLC, author = "Ville Meril{\"a}", title = "A Nonvanishing Lemma for Certain {Pad{\'e}} Approximations of the Second Kind", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "1977--1997", month = dec, year = "2011", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042111004964", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004964", abstract = "We prove the nonvanishing lemma for explicit second kind Pad{\'e} approximations to generalized hypergeometric and q-hypergeometric functions. The proof is based on an evaluation of a generalized Vandermonde determinant. Also, some immediate applications to the Diophantine approximation is given in the form of sharp linear independence measures for hypergeometric E- and G-functions in algebraic number fields with different valuations.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Nathanson:2011:PAN, author = "Melvyn B. Nathanson", title = "Problems in additive number theory, {IV}: Nets in groups and shortest length $g$-adic representations", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "1999--2017", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004940", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004940", abstract = "The number theoretic analog of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer $ g \geq 2 $, the study of $h$-nets in the additive group of integers with respect to the generating set $ A_g = \{ 0 \} \cup \{ \pm g^i \colon i = 0, 1, 2, \ldots \} $ requires a knowledge of the word lengths of integers with respect to $ A_g$. A $g$-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Haessig:2011:FSP, author = "C. Douglas Haessig and Antonio Rojas-Le{\'o}n", title = "{$L$}-Functions of Symmetric Powers of the Generalized {Airy} Family of Exponential Sums", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2019--2064", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005040", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005040", abstract = "This paper looks at the {$L$}-function of the kth symmetric power of the -sheaf Ai$_f$ over the affine line associated to the generalized Airy family of exponential sums. Using \ell -adic techniques, we compute the degree of this rational function as well as the local factors at infinity. Using $p$-adic techniques, we study the $q$-adic Newton polygon of the {$L$}-function.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Pande:2011:DGR, author = "Aftab Pande", title = "Deformations of {Galois} Representations and the Theorems of {Sato--Tate} and {Lang--Trotter}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2065--2079", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004939", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004939", abstract = "We construct infinitely ramified Galois representations \rho such that the a$_l$ (\rho)'s have distributions in contrast to the statements of Sato--Tate, Lang--Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalizing a result of Ramakrishna.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Bremner:2011:X, author = "Andrew Bremner and Maciej Ulas", title = "On $ x^a \pm y^b \pm z^c \pm w^d = 0 $, $ 1 / a + 1 / b + 1 / c + 1 / d = 1 $", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2081--2090", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005076", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005076", abstract = "It is well known that the Diophantine equations x$^4$ + y$^4$ = z$^4$ + w$^4$ and x$^4$ + y$^4$ + z$^4$ = w$^4$ each have infinitely many rational solutions. It is also known for the equation x$^6$ + y$^6$- z$^6$ = w$^2$. We extend the investigation to equations x$^a$ \pm y$^b$ = \pm z$^c$ \pm w$^d$, a, b, c, d \in Z, with 1/a + 1/b + 1/c + 1/d = 1. We show, with one possible exception, that if there is a solution of the equation in the reals, then the equation has infinitely many solutions in the integers. Of particular interest is the equation x$^6$ + y$^6$ + z$^6$ = w$^2$ because of its classical nature; but there seem to be no references in the literature.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Yu:2011:EMO, author = "Chia-Fu Yu", title = "On the Existence of Maximal Orders", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2091--2114", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005003", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005003", abstract = "We generalize the existence of maximal orders in a semi-simple algebra for general ground rings. We also improve several statements in Chaps. 5 and 6 of Reiner's book [ {\em Maximal Orders\/}, London Mathematical Society Monographs, Vol. 5 (Academic Press, London, 1975), 395 pp.] concerning separable algebras by removing the separability condition, provided the ground ring is only assumed to be Japanese, a very mild condition. Finally, we show the existence of maximal orders as endomorphism rings of abelian varieties in each isogeny class.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Qi:2011:MTS, author = "Zhi Qi and Chang Yang", title = "{Morita}'s Theory for the Symplectic Groups", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2115--2137", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004952", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004952", abstract = "We construct and study the holomorphic discrete series representations and the principal series representations of the symplectic group Sp(2n, F) over a $p$-adic field F as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for SL(2, F) defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Lebacque:2011:LDZ, author = "Philippe Lebacque and Alexey Zykin", title = "On Logarithmic Derivatives of Zeta Functions in Families of Global Fields", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2139--2156", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005015", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005015", abstract = "We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer--Siegel theorem both for number fields and function fields.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Habsieger:2011:SCP, author = "Laurent Habsieger and Emmanuel Royer", title = "Spiegelungssatz: a Combinatorial Proof for the $4$-Rank", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2157--2170", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005106", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005106", abstract = "The Spiegelungssatz is an inequality between the 4-ranks of the narrow ideal class groups of the quadratic fields and . We provide a combinatorial proof of this inequality. Our interpretation gives an affine system of equations that allows to describe precisely some equality cases.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Vega:2011:HFF, author = "M. Valentina Vega", title = "Hypergeometric Functions Over Finite Fields and Their Relations to Algebraic Curves", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2171--2195", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004976", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004976", abstract = "In this work we present an explicit relation between the number of points on a family of algebraic curves over {$ \mathbb {F} $}$_q$ and sums of values of certain hypergeometric functions over {$ \mathbb {F} $}$_q$. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over {$ \mathbb {F} $}$_q$ in some particular cases. A general conjecture relating these last two is presented and advances toward its proof are shown in the last section.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Jabuka:2011:WTD, author = "Stanislav Jabuka and Sinai Robins and Xinli Wang", title = "When Are Two {Dedekind} Sums Equal?", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2197--2202", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005088", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005088", abstract = "A natural question about Dedekind sums is to find conditions on the integers a$_1$, a$_2$, and b such that s(a$_1$, b) = s(a$_2$, b). We prove that if the former equality holds then b|(a$_1$ a$_2$- 1)(a$_1$- a$_2$). Surprisingly, to the best of our knowledge such statements have not appeared in the literature. A similar theorem is proved for the more general Dedekind--Rademacher sums as well, namely that for any fixed non-negative integer n, a positive integer modulus b, and two integers a$_1$ and a$_2$ that are relatively prime to b, the hypothesis r$_n$ (a$_1$, b) = r$_n$ (a$_2$, b) implies that b|(6n$^2$ + 1 - a$_1$ a$_2$)(a$_2$- a$_1$).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Smith:2011:VBD, author = "Ethan Smith", title = "A Variant of the {Barban--Davenport--Halberstam Theorem}", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2203--2218", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S179304211100499X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100499X", abstract = "Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals {$ \mathfrak {p} $} of K with fixed Frobenius class in Gal(L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotar{\"e}v Density Theorem for this problem is small ``on average''. The result may be viewed as a variation on the classical Barban--Davenport--Halberstam Theorem.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Harvey:2011:CDI, author = "M. P. Harvey", title = "Cubic {Diophantine} Inequalities Involving a Norm Form", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2219--2235", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005052", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005052", abstract = "We apply Freeman's variant of the Davenport--Heilbronn method to provide an asymptotic formula for the number of small values taken by a certain family of cubic forms with real coefficients. The cubic forms in question arise as the sum of a diagonal form and a norm form and should have at least seven variables.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Rolen:2011:GCN, author = "Larry Rolen", title = "A Generalization of the Congruent Number Problem", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2237--2247", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005039", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005039", abstract = "We study a certain generalization of the classical Congruent Number Problem. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle \theta. These numbers are called \theta -congruent. We give an elliptic curve criterion for determining whether a given integer n is \theta congruent. We then consider the ``density'' of integers n which are \theta -congruent, as well as the related problem giving the ``density'' of angles \theta for which a fixed n is congruent. Assuming the Shafarevich--Tate conjecture, we prove that both proportions are at least 50\% in the limit. To obtain our result we use the recently proven $p$-parity conjecture due to Monsky and the Dokchitsers as well as a theorem of Helfgott on average root numbers in algebraic families.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Radu:2011:CPM, author = "Silviu Radu and James A. Sellers", title = "Congruence properties modulo $5$ and $7$ for the {$ \mathrm {pod}$} function", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2249--2259", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005064", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005064", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{ElBachraoui:2011:IFS, author = "Mohamed {El Bachraoui}", title = "Inductive Formulas for Some Arithmetic Functions", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2261--2268", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S179304211100509X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211100509X", abstract = "We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Graves:2011:NPE, author = "Hester Graves", title = "Has a Non-Principal {Euclidean} Ideal", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2269--2271", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111004988", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111004988", abstract = "This paper introduces a totally real quartic number field with a non-principal Euclidean ideal.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kim:2011:AVM, author = "Min-Soo Kim and Su Hu", title = "A $p$-adic view of multiple sums of powers", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2273--2288", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005027", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005027", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Anonymous:2011:AIV, author = "Anonymous", title = "Author Index (Volume 7)", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "8", pages = "2289--2295", month = dec, year = "2011", DOI = "https://doi.org/10.1142/S1793042111005118", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111005118", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{FreixasIMontplet:2012:JLC, author = "Gerard {Freixas I.Montplet}", title = "The {Jacquet--Langlands} Correspondence and the Arithmetic {Riemann--Roch} Theorem for Pointed Curves", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "1--29", month = feb, year = "2012", CODEN = "????", DOI = "https://doi.org/10.1142/S1793042112500017", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500017", abstract = "We show how the Jacquet--Langlands correspondence and the arithmetic Riemann--Roch theorem for pointed curves, relate the arithmetic self-intersection numbers of the sheaves of modular forms with their Petersson norms --- on modular and Shimura curves: these are equal modulo $ \sum_{l \in S} $ {$ \mathbb {Q} $} log l, where S is a controlled set of primes. These quantities were previously considered by Bost and K{\"u}hn (modular curve case) and Kudla--Rapoport--Yang and Maillot--Roessler (Shimura curve case). By the work of Maillot and Roessler, our result settles a question raised by Soul{\'e}.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Finotti:2012:NPC, author = "Lu{\'i}s R. A. Finotti", title = "Nonexistence of Pseudo-Canonical Liftings", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "31--51", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500029", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500029", abstract = "In this paper we show that pseudo-canonical liftings do not exist, by showing that if j$_0$ \mapsto (j$_0$, J$_1$ (j$_0$), J$_2$ (j$_0$),\ldots) is the map that gives canonical liftings for ordinary j$_0$, then J$_2$ has a pole at j$_0$ = 1728 if p \equiv 3 (mod 4) and J$_3$ has a pole at j$_0$ = 0 if p \equiv 5 (mod 6). Moreover, precise descriptions of J$_2$ and J$_3$ are given.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Akbary:2012:GVT, author = "Amir Akbary and Dragos Ghioca", title = "A Geometric Variant of {Titchmarsh} Divisor Problem", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "53--69", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500030", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500030", abstract = "We formulate a geometric analog of the Titchmarsh divisor problem in the context of abelian varieties. For any abelian variety A defined over {$ \mathbb {Q} $}, we study the asymptotic distribution of the primes of {\mathbb{Z}} which split completely in the division fields of A. For all abelian varieties which contain an elliptic curve we establish an asymptotic formula for such primes under the assumption of Generalized Riemann Hypothesis. We explain how to derive an unconditional asymptotic formula in the case that the abelian variety is a complex multiplication elliptic curve.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Amdeberhan:2012:II, author = "Tewodros Amdeberhan and Christoph Koutschan and Victor H. Moll and Eric S. Rowland", title = "The iterated integrals of $ \ln (1 + x^n) $", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "71--94", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500042", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500042", abstract = "For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x$^2$), arithmetic properties of certain coefficients arising are described. Similar observations are made for ln(1 + x$^3$).", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Castillo:2012:HOS, author = "Daniel Macias Castillo", title = "On higher-order {Stickelberger}-type theorems for multi-quadratic extensions", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "95--110", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500054", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500054", abstract = "We prove, for all quadratic and a wide range of multi-quadratic extensions of global fields, a result concerning the annihilation as Galois modules of ideal class groups by explicit elements constructed from the values of higher-order derivatives of Dirichlet {$L$}-functions. This result simultaneously refines Rubin's integral version of Stark's Conjecture and provides evidence for the relevant case of the Equivariant Tamagawa Number Conjecture of Burns and Flach.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Chan:2012:TCA, author = "Song Heng Chan and Renrong Mao", title = "Two Congruences for {Appell--Lerch} Sums", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "111--123", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500066", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500066", abstract = "Two congruences are proved for an infinite family of Appell--Lerch sums. As corollaries, special cases imply congruences for some of the mock theta functions of order two, six and eight.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kadiri:2012:EZF, author = "Habiba Kadiri", title = "Explicit Zero-Free Regions for {Dedekind} Zeta Functions", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "125--147", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500078", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500078", abstract = "Let K be a number field, n$_K$ be its degree, and d$_K$ be the absolute value of its discriminant. We prove that, if d$_K$ is sufficiently large, then the Dedekind zeta function \zeta$_K$ (s) has no zeros in the region:, , where log M = 12.55 log d$_K$ + 9.69n$_K$ log|\Im {$ \mathfrak {m} $} s| + 3.03 n$_K$ + 58.63. Moreover, it has at most one zero in the region:, . This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: \zeta$_K$ (s) has at most one zero in the region, .", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Schweiger:2012:DAR, author = "F. Schweiger", title = "A $2$-Dimensional Algorithm Related to the {Farey--Brocot} Sequence", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "149--160", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S179304211250008X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211250008X", abstract = "Moshchevitin and Vielhaber gave an interesting generalization of the Farey--Brocot sequence for dimension d \geq 2 (see [N. Moshchevitin and M. Vielhaber, Moments for generalized Farey--Brocot partitions, {\em Funct. Approx. Comment. Math.\/} 38 (2008), part 2, 131--157]). For dimension d = 2 they investigate two special cases called algorithm and algorithm. Algorithm is related to a proposal of M{\"o}nkemeyer and to Selmer algorithm (see [G. Panti, Multidimensional continued fractions and a Minkowski function, {\em Monatsh. Math.\/} 154 (2008) 247--264]). However, algorithm seems to be related to a new type of 2-dimensional continued fractions. The content of this paper is first to describe such an algorithm and to give some of its ergodic properties. In the second part the dual algorithm is considered which behaves similar to the Parry--Daniels map.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Filaseta:2012:S, author = "M. Filaseta and S. Laishram and N. Saradha", title = "Solving $ n (n + d) \cdots (n + (k - 1) d) = b y^2 $ with {$ P(b) \leq C k $}", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "161--173", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500091", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500091", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Noble:2012:AWD, author = "Rob Noble", title = "Asymptotics of the Weighted {Delannoy} Numbers", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "175--188", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500108", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500108", abstract = "The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this paper we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Fukshansky:2012:WRI, author = "Lenny Fukshansky and Kathleen Petersen", title = "On Well-Rounded Ideal Lattices", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "1", pages = "189--206", month = feb, year = "2012", DOI = "https://doi.org/10.1142/S179304211250011X", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:27 MDT 2020", bibsource = "http://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S179304211250011X", abstract = "We investigate a connection between two important classes of Euclidean lattices: we