%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.03", %%% date = "24 June 2022", %%% time = "11:36:33 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 = "22945 48665 217357 2191546", %%% 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 1.03, 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 ( 131) %%% 2010 ( 111) 2016 ( 143) 2022 ( 95) %%% %%% Article: 1962 %%% %%% Total entries: 1962 %%% %%% 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", 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", 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",