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%%% ====================================================================
%%%  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
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%%%
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%%%                        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''.
%%%
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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",