Functiones et Approximatio Commentarii Mathematici Articles (Project Euclid)

On some diophantine results related to Hermite polynomials

Thu, 05 Aug 2010 15:41 EDT

Csaba Rakaczki

Source: Funct. Approx. Comment. Math., Volume 42, Number 1, 7--16.

Abstract:
In this paper we prove that the shifted Hermite polynomial $H_{n}(x)+b$ has at least three simple zeros for each complex number $b$, provided that $n\geq 7$.

Units in real cyclic fields

Mon, 26 Sep 2011 09:53 EDT

Roman Marszałek

Source: Funct. Approx. Comment. Math., Volume 45, Number 1, 139--153.

Abstract:
Let $N/\mathbb{Q}$ be a real cyclic and tame extension of prime degree $l$ with $\Gamma=\mathscr{G}al(N/\mathbb{Q})$. We give the Hom description of the class of the torsion-free part of the group of units in $N$ in the class group of the order $\mathbb{Z}\Gamma/ (\sum_{\gamma \in \Gamma}\gamma )$. This representation depends only on the structure of the ideal class group of $N$ and determines the Galois module structure of the torsion-free part of the group of units in $N$ as an ideal of the $l$th cyclotomic field. Using this approach we derive necessary and sufficient conditions for all real and tame cyclic fields of prime degree to have Minkowski units. We extend also the class of known cyclic real fields with Minkowski units.

On multiple exponential sums and their applications

Mon, 12 Dec 2011 11:03 EST

H.-Q Liu

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 155--163.

Abstract:
We prove new estimates for the remainder terms in the known asymptotic formulas for three famous problems, by using the contemporary bounds for triple exponential sums.

Fleck's congruence, associated magic squares and a zeta identity

Mon, 12 Dec 2011 11:03 EST

Matthew C. Lettington

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 165--205.

Abstract:
Let the \emph{Fleck numbers}, $C_n(t,q)$, be defined such that \[ C_n(t,q)=\sum_{k\equiv q (mod n)}(-1)^k\binom{t}{k}. \] For prime $p$, Fleck obtained the result $C_p(t,q)\equiv 0 (mod p^{{\left \lfloor (t-1)/(p-1)\right \rfloor}}} )$, where $\lfloor.\rfloor$ denotes the usual floor function. This congruence was extended 64 years later by Weisman, in 1977, to include the case $n=p^\alpha$. In this paper we show that the Fleck numbers occur naturally when one considers a symmetric $n\times n$ matrix, $M$, and its inverse under matrix multiplication. More specifically, we take $M$ to be a symmetrically constructed $n\times n$ associated magic square of odd order, and then consider the reduced coefficients of the linear expansions of the entries of $M^t$ with $t\in \mathbb{Z}$. We also show that for any odd integer, $n=2m+1$, $n\geq 3$, there exist geometric polynomials in $m$ that are linked to the Fleck numbers via matrix algebra and $p$-adic interaction. These polynomials generate numbers that obey a reciprocal type of congruence to the one discovered by Fleck. As a by-product of our investigations we observe a new identity between values of the Zeta functions at even integers. Namely \[ \zeta{(2j)}=(-1)^{j+1}\left (\frac{j\pi^{2j}}{(2j+1)!}+\sum_{k=1}^{j-1}\frac{(-1)^k\pi^{2j-2k}}{(2j-2k+1)!}\zeta{(2k)}\right ). \] We conclude with examples of combinatorial congruences, Vandermonde type determinants and Number Walls that further highlight the symmetric relations that exist between the Fleck numbers and the geometric polynomials.

Jeśmanowicz' conjecture on exponential diophantine equations

Mon, 12 Dec 2011 11:03 EST

Takafumi Miyazaki

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 207--229.

Abstract:
Let $(a,b,c)$ be a primitive Pythagorean triple such that $a^2+b^2=c^2$ with even $b$. In 1956, L. Jeśmanowicz conjectured that the equation $a^x+b^y=c^z$ has only the solution $(x,y,z)=(2,2,2)$ in positive integers. In this paper, we give various new results on this conjecture. In particular, we prove that if the equation has a solution $(x,y,z)$ with even $x,z$ then $x/2$ and $z/2$ are odd.

Construction of normal numbers by classified prime divisors of integers

Mon, 12 Dec 2011 11:03 EST

Jean-Marie De Koninck, Imre Kátai

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 231--253.

Abstract:
Given an integer $d\ge 2$, a $d$-{\it normal number}, or simply a {\it normal number}, is a real number whose $d$-ary expansion is such that any preassigned sequence, of length $k\ge 1$, of base $d$ digits from this expansion, occurs at the expected frequency, namely $1/d^k$. We construct large families of normal numbers using classified prime divisors of integers.

Some conditional results on primes between consecutive squares

Mon, 12 Dec 2011 11:03 EST

Danilo Bazzanella

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 255--263.

Abstract:
A well-known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide the upper bounds for the exceptional set for this conjecture under the assumption of some heuristic hypotheses.

Arbitrary potential modularity for elliptic curves over totally real number fields

Mon, 12 Dec 2011 11:03 EST

Cristian Virdol

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 265--269.

Abstract:
In this paper we prove the arbitrary potential modularity for an elliptic curve defined over a totally real number field.

Euler-Rabinowitsch polynomials and class number problems revisited

Mon, 12 Dec 2011 11:03 EST

Richard A. Mollin, Anitha Srinivasan

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 271--288.

Abstract:
We prove a conjecture posed in [11] and continue the process of determining Euler-Rabinowitsch polynomials that produce consecutive primes in a given range of inputs, and the relationship with class numbers of the underlying quadratic field.

Hermite's formulas for $q$-analogues of Hurwitz zeta functions

Mon, 12 Dec 2011 11:03 EST

Yoshinobu Tomita

Source: Funct. Approx. Comment. Math., Volume 45, Number 2, 289--301.

Abstract:
We treat Hermite's formulas for $q$-analogues of the Hurwitz zeta function. As their application, we study the classical limit of modified $q$-analogues of the Hurwitz zeta function. We also treat $q$-analogues of the Milnor multiple gamma function.

Powers in $\prod\limits_{k=1}^n (ak^{2^l\cdot3^m}+b)$

Fri, 30 Mar 2012 09:09 EDT

Zhongfeng Zhang

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 7--13.

Abstract:
Let $f(x)=ax^{2^l\cdot3^m}+b\in \mathbb{Z}[x]$ be a polynomial with $l\geq 1, l+m\geq 2, ab\neq 0$ and such that $f(k)\neq 0$ for any $k\geq 1$. We prove, under $ABC$ conjecture, that the product $\prod_{k=1}^n f(k)$ is not a $2^l\cdot3^m$-th power for $n$ large enough.

On Cauchy-Liouville-Mirimanoff polynomials II

Fri, 30 Mar 2012 09:09 EDT

Pavlos Tzermias

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 15--27.

Abstract:
The main result of [23] on the non-existence of low-degree irreducible factors of the Cauchy-Liouville-Mirimanoff polynomials $E_p(x)$ for primes $p \equiv 2 \pmod{3}$ is extended to a similar result for $p \equiv 1 \pmod{3}$. We also give a partial result on the existence of higher-degree irreducible factors.

New identities for Ramanujan's cubic continued fraction

Fri, 30 Mar 2012 09:09 EDT

M.S. Mahadeva Naika, S. Chandankumar, K. Sushan Bairy

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 29--44.

Abstract:
In this paper, we present some new identities providing relations between Ramanujan's cubic continued fraction $V(q)$ and the other three continued fractions $V(q^9)$, $V(q^{17})$ and $V(q^{19})$. In the process, we establish some new modular equations for the ratios of Ramanujan's theta functions. We also establish some general formulas for the explicit evaluations of ratios of Ramanujan's theta functions.

Multiple polylogarithms and multi-poly-Bernoulli polynomials

Fri, 30 Mar 2012 09:09 EDT

Abdelmejid Bayad, Yoshinori Hamahata

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 45--61.

Abstract:
In this paper we introduce special generalized Bernoulli polynomials which generalize poly-Bernoulli polynomials and numbers. We call them multi-poly-Bernoulli polynomials and numbers. We prove a collection of important and fundamental identities satisfied by our multi-poly-Bernoulli polynomials and numbers.

Evaluations of some quadruple Euler sums of even weight

Fri, 30 Mar 2012 09:09 EDT

Minking Eie, Chuan-Sheng Wei

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 63--77.

Abstract:
For positive integers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}$ with $\alpha_{r} \geq 2$, the multiple zeta value or $r$-fold Euler sum is defined by \[\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}) = \sum_{1\leq n_{1}<n_{2}<\cdots<n_{r}}n_1^{-\alpha_1}n_2^{-\alpha_2}\cdots n_r^{-\alpha_r}. \] where $|\alpha|=\alpha_1+\alpha_2+\cdots+\alpha_r$ and $r$ are the weight and depth of $\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$ respectively. By the general theorem given in [11], the multiple zeta value $\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$ can be expressed as a rational linear combination of products of multiple zeta values of lower depth if its depth and weight are of different parity. In other words, when the sum of its depth and weight is odd. However, there are still some exceptions for quadruple Euler sums. As conjectured in [6], a quadruple Euler sum with even weight exceeding 14 can be expressed as a rational linear combination of products of multiple zeta values of depth 1, 2 and 3 if and only if it is one of the following forms: $\zeta(1,a,b,a)$, $\zeta(b,1,a,a)$, $\zeta(b,b,1,a)$ or $\zeta(a,b,b,a)$ with $a=b$ or $b=1$. In this paper, we shall evaluate these quadruple Euler sums of even weight by the identities among multiple zeta values with variables and relation obtained from the stuffle formula of two multiple zeta values.

A combinatorial-geometric viewpoint of Knopp's formula for Dedekind sums

Fri, 30 Mar 2012 09:09 EDT

Kazuhito Kozuka

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 79--89.

Abstract:
In this paper, by means of a combinatorial-geometric method, we give a new proof of Knopp's formula for Dedekind sums and its generalizations to multiple Dedekind sums attached to Dirichlet characters. The combinatorial-geometric method for studying Dedekind sums were introduced by Beck, who proved the well-known reciprocity formula for Dedekind sums and some of its generalizations by the method. The motive of this paper is to find a similar approch to Knopp's formula .

On Weyl sums for smaller exponents

Fri, 30 Mar 2012 09:09 EDT

Kent D. Boklan, Trevor D. Wooley

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 91--107.

Abstract:
We present a hybrid approach to bounding exponential sums over $k$th powers via Vinogradov's mean value theorem, and derive estimates of utility for exponents $k$ of intermediate size.

On the diophantine equation $2^x=x^2+y^2-2$

Fri, 30 Mar 2012 09:09 EDT

Alexandru Gica, Florian Luca

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 109--116.

Abstract:
In this paper, we show that the only positive integer solutions of the equation $2^x=x^2+y^2-2$ are $(x,y)=(3,1),~(5,3),~(7,9)$. We propose also the following conjecture: the equation $2^x=y^2+z^2(x^2-2)$, where $y,z$ are odd positive integers and $x$ is a positive integer such that $x^2-2$ is a prime number, has the only solutions $(x,y,z)=(3,1,1),~(5,3,1),~(7,9,1),~(13,3,7)$. The conjecture implies a recent result of Lee [4] which states that if $x^2-2$ is an odd prime number such that the class number $h(x^2-2)$ of the quadratic field $\mathbb{Q}[{\sqrt{x^2-2}}]$ is $1$, then $x=3,5,7,13$.

Minimal genus one curves

Fri, 30 Mar 2012 09:09 EDT

Mohammad Sadek

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 117--131.

Abstract:
In this paper we consider genus one equations of degree $n$, namely a (generalised) binary quartic when $n=2$, a ternary cubic when $n=3$, and a pair of quaternary quadrics when $n=4$. A new definition for the minimality of genus one equations of degree $n$ over local fields is introduced. The advantage of this definition is that it does not depend on invariant theory of genus one curves. We prove that this definition coincides with the classical definition of minimality for all $n\le4$. As an application, we give a new proof for the existence of global minimal genus one equations over number fields of class number 1.

Perfect powers generated by the twisted Fermat cubic

Fri, 30 Mar 2012 09:09 EDT

Jonathan Reynolds

Source: Funct. Approx. Comment. Math., Volume 46, Number 1, 133--145.

Abstract:
On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than $1$. Moreover, if the first term is divisible by $6$ and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an $l$th power for some $l$ dividing the order of $2$ in the first term.

Fourier coefficients of Hecke eigenforms

Mon, 25 Jun 2012 08:47 EDT

Ronald Evans

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 147--159.

Abstract:
We provide systematic evaluations, in terms of binary quadratic representations of $4p$, for the $p$-th Fourier coefficients of each member $f$ of an infinite class $\mathcal{C}$ of CM eigenforms. As an application, previously conjectured evaluations of three algebro-geometric character sums can now be formulated explicitly without reference to eigenforms. There are several non-CM newforms that appear to share some properties with the eigenforms in $\mathcal{C}$, and we pose some conjectures about their Fourier coefficients.

Periodicity of complementing multisets

Mon, 25 Jun 2012 08:47 EDT

Željka Ljujić

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 161--175.

Abstract:
Let $A$ be a finite multiset of integers. If $B$ be a multiset such that $A$ and $B$ are $t$-complementing multisets of integers, then $B$ is periodic. We obtain the Biro-type upper bound for the smallest such period of $B$: Let $\varepsilon>0$. We assume that $diam(A)\ge n_0(\varepsilon)$ and that $\sum_{a\in A}w_A(a)\leq (diam(A)+1)^{c}$, where $c$ is any constant such that $c< 100\log2-2$. Then $B$ is periodic with period $\log k\leq (diam(A)+1)^{\frac{1}{3}+\varepsilon}$.

A note on a conjecture of Gonek

Mon, 25 Jun 2012 08:47 EDT

Micah B. Milinovich, Nathan Ng

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 177--187.

Abstract:
We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function over the zeros of $\zeta(s)$ that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.

Base change and the Birch and Swinnerton-Dyer conjecture

Mon, 25 Jun 2012 08:47 EDT

Cristian Virdol

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 189--194.

Abstract:
In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for products of abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for products of abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.

How slowly can a bounded sequence cluster?

Mon, 25 Jun 2012 08:47 EDT

John Bentin

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 195--204.

Abstract:
We propose a simple measure of how slowly a bounded real sequence clusters. This measure, called \emph{separation}, is the infimum, over all finite segments of the sequence with at least two terms, of the ratio of the least distance between the terms in the segment to the general supremum of such a distance for a segment of that length. An example of a highly separated sequence is given. To create a more separated sequence, we modify van der Corput's construction, replacing the powers of a base by the even-numbered terms of the Fibonacci sequence. The result coincides initially with the sequence built stepwise by maximizing separation for each extra term. We conjecture that these sequences are the same and of maximal separation.

The absolute Galois group of subfields of the field of totally $\boldsymbol{S}$-adic numbers

Mon, 25 Jun 2012 08:47 EDT

Dan Haran, Moshe Jarden, Florian Pop

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 205--223.

Abstract:
For a finite set $S$ of local primes of a countable Hilbertian field $K$ and for $\sigma_1,\ldots,\sigma_e\in\Gal(K)$ we denote the field of totally $S$-adic numbers by $\K_{tot,S}$, the fixed field of $\sigma_1,\ldots,\sigma_e$ in $\K_{tot,S}$ by $\K_{tot,S}({\mathbf \sigma})$, and the maximal Galois extension of $K$ in $\KtotS({\mathbf \sigma})$ by $\KtotS[{\mathbf \sigma}]$. We prove that for almost all ${\mathbf \sigma}\in\Gal(K)^e$ the absolute Galois group of $\K_{tot,S}[{\mathbf \sigma}]$ is isomorphic to the free product of $\hat{F}_\omega$ and a free product of local factors over $S$.

A PNT equivalence for Beurling numbers

Mon, 25 Jun 2012 08:47 EDT

Harold G. Diamond, Wen-Bin Zhang

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 225--234.

Abstract:
In classical prime number theory, several relations are considered to be equivalent to the Prime Number Theorem. For Beurling generalized numbers, some auxiliary conditions may be needed to deduce one relation from another one. We show conditions under which the Beurling analog of the sharp version of Mertens' sum formula does or does not hold.

Divisor functions over quaternion algebras\newline and a type of identities

Mon, 25 Jun 2012 08:47 EDT

Yichao Zhang

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 235--260.

Abstract:
In order to prove a result on fourth moments of modular L-functions, Duke derived an identity of the divisor function over the rational Hamiltonian quaternion algebra. Recently Kim and the author generalized Duke's divisor function, the identity and other related results from level two to general prime level. In this note, we consider such identities in general.

Some arithmetic identities involving divisor functions

Mon, 25 Jun 2012 08:47 EDT

Şaban Alaca, Faruk Uygul, Kenneth S. Williams

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 261--271.

Abstract:
For a positive integer $n$, let $\sigma(n):= \sum_{d \in \mathb{N}, d|n} d$. The explicit evaluation of such arithmetic sums as $\sum_{(a,b,c) \in \ABIFnn^3, a+2b+4c=n} \sigma(a)\sigma(b) \sigma(c)$ and $\sum_{(a,b) \in \ABIFnn^2, a+2b=n} a \sigma(a)\sigma(b)$ is carried out for all positive integers $n$.

On the constant in Burgess' bound for the number of consecutive residues or non-residues

Mon, 25 Jun 2012 08:47 EDT

Kevin J. McGown

Source: Funct. Approx. Comment. Math., Volume 46, Number 2, 273--284.

Abstract:
We give an explicit version of a result due to D. Burgess. Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $\chi$ takes on a particular value is less than $\left\{\frac{\pi e\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ term is given explicitly.

On the critical values of $L$-functions of base change for Hilbert modular forms II

Tue, 25 Sep 2012 09:05 EDT

Cristian Virdol

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 7--13.

Abstract:
In this paper we generalize some results, obtained by Shimura, Yoshida and the author, on critical values of $L$-functions of $l$-adic representations attached to Hilbert modular forms twisted by finite order characters, to the critical values of $L$-functions of arbitrary base change to totally real number fields of $l$-adic representations attached to Hilbert modular forms twisted by some finite-dimensional representations.

Power series with the von Mangoldt function

Tue, 25 Sep 2012 09:05 EDT

Matthias Kunik, Lutz G Lucht

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 15--33.

Abstract:
We study the analytic behavior of a power series with coefficients containing the von Mangoldt function. In particular, we extend an explicit formula of Hardy and Littlewood for related functions and derive further representation formulas in the unit disk that reveal logarithmic singularities on a dense subset of the unit circle. As an essential tool for proving the square integrability of occurring limit functions together with respective error estimates we contribute a new proof of a Ramanujan-like expansion of an arithmetic function consisting of the von Mangoldt function and the Euler function.

On the simplest sextic fields and related Thue equations

Tue, 25 Sep 2012 09:05 EDT

Akinari Hoshi

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 35--49.

Abstract:
We consider the parametric family of sextic Thue equations $$x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda$$ where $m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.

Irreducibility of generalized Hermite-Laguerre polynomials

Tue, 25 Sep 2012 09:05 EDT

Shanta Laishram, Tarlok N Shorey

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 51--64.

Abstract:
For a rational $q=u+\frac{\alpha}{d}$ with $u, \alpha, d\in \ACOBZ$ with $u\ge 0, 1\le \alpha<d$, $\gcd(\alpha, d)=1$, the \emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \begin{align*} G_q(x)&=a_nx^n+a_{n-1}(\alpha +(n-1+u)d)x^{n-1}+\cdots\\ &\quad+a_1\left(\prod^{n-1}_{i=1}(\alpha +(i+u)d)\right)x+a_0 \left(\prod^{n-1}_{i=0}(\alpha +(i+u)d)\right) \end{align*} where $a_0, a_1, \cdots , a_n$ are arbitrary integers. We prove some irreducibility results of $G_q(x)$ when $q\in \{\frac{1}{3}, \frac{2}{3}\}$ and extend some of the earlier irreducibility results when $q$ of the form $u+\frac{1}{2}$. We also prove a new improved lower bound for greatest prime factor of product of consecutive terms of an arithmetic progression whose common difference is $2$ and $3$.

Poisson type phenomena for points on hyperelliptic curves modulo $p$

Tue, 25 Sep 2012 09:05 EDT

Kit-Ho Mak, Alexandru Zaharescu

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 65--78.

Abstract:
Let $p$ be a large prime, and let $C$ be a hyperelliptic curve over $\mathbb{F}_p$. We study the distribution of the $x$-coordinates in short intervals when the $y$-coordinates lie in a prescribed interval, and the distribution of the distance between consecutive $x$-coordinates with the same property. Next, let $g(P,P_0)$ be a rational function of two points on $C$. We study the distribution of the above distances with an extra condition that $g(P_i,P_{i+1})$ lies in a prescribed interval, for any consecutive points $P_i,P_{i+1}$.

Zeros of the derivatives of the Riemann zeta-function

Tue, 25 Sep 2012 09:05 EDT

Haseo Ki, Yoonbok Lee

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 79--87.

Abstract:
Levinson and Montgomery in 1974 proved many interesting formulae on the zeros of derivatives of the Riemann zeta function $\zeta(s)$. When Conrey proved that at least 2/5 of the zeros of the Riemann zeta function are on the critical line, he proved the asymptotic formula for the mean square of $\zeta(s)$ multiplied by a mollifier of length $ T^{4/7}$ near the $1/2$-line. As a consequence of their papers, we study some aspects of zeros of the derivatives of the Riemann zeta function with no assumption.

On torsion points of certain CM elliptic curves

Tue, 25 Sep 2012 09:05 EDT

Naoki Murabayashi

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 89--93.

Abstract:
Let $E$ be a CM elliptic curve defined over an algebraic number field $F$ with CM by an imaginary quadratic field $K$. We determine the group of $K_{ab}F$-rational torsion points of $E$. In some cases we also determine the group of $F$ or $KF$-rational torsion points of $E$.

On $\lambda$-invariants of $\mathbb{Z}_\ell$-extensions over real abelian number fields of prime power conductors

Tue, 25 Sep 2012 09:05 EDT

Takashi Fukuda, Keiichi Komatsu, Takayuki Morisawa

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 95--104.

Abstract:
For each prime number $\ell$ less than $10^4$, we construct an infinite family of abelian number fields for which Iwasawa $\lambda_{\ell}$-invariants vanish.

Height reducing problem on algebraic integers

Tue, 25 Sep 2012 09:05 EDT

Shigeki Akiyama, Paulius Drungilas, Jonas Jankauskas

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 105--119.

Abstract:
Let $\alpha$ be an algebraic integer and assume that it is {\it expanding}, i.e., its all conjugates lie outside the unit circle. We show several results of the form $\mathbb{Z}[\alpha]=\mathcal{B}[\alpha]$ with a certain finite set $\mathcal{B}\subset\mathbb{Z}$. This property is called {\it height reducing property}, which attracted special interest in the self-affine tilings. Especially we show that if $\alpha$ is quadratic or cubic trinomial, then one can choose $\mathcal{B}= \left\{0,\,\pm 1,\,\ldots,\,\pm \left(|N(\alpha)|-1\right)\right\}$, where $N(\alpha)$ stands for the absolute norm of $\\alpha$ over $\mathbb{Q}$.

Algebraic independence of certain numbers related to modular functions

Tue, 25 Sep 2012 09:05 EDT

Carsten Elsner, Shun Shimomura, Iekata Shiokawa

Source: Funct. Approx. Comment. Math., Volume 47, Number 1, 121--141.

Abstract:
In previous papers the authors established a method how to decide on the algebraic independence of a set $\{ y_1,\dots ,y_n \}$ when these numbers are connected with a set $\{ x_1,\dots ,x_n \}$ of algebraic independent parameters by a system $f_i(x_1,\dots ,x_n,y_1,\dots ,y_n) =0$ $(i=1,2,\dots ,n)$ of rational functions. Constructing algebraic independent parameters by Nesterenko's theorem, the authors successfully applied their method to reciprocal sums of Fibonacci numbers and determined all the algebraic relations between three $q$-series belonging to one of the sixteen families of $q$-series introduced by Ramanujan. In this paper we first give a short proof of Nesterenko's theorem on the algebraic independence of $\pi$, $e^{\pi\sqrt{d}}$ and a product of Gamma-values $\Gamma (m/n)$ at rational points $m/n$. Then we apply the method mentioned above to various sets of numbers. Our algebraic independence results include among others the coefficients of the series expansion of the Heuman-Lambda function, the values $P(q^r), Q(q^r)$, and $R(q^r)$ of the Ramanujan functions $P,Q$, and $R$, for $q\in \overline{\ACADQ}$ with $0<|q|<1$ and $r=1,2,3,5,7,10$, and the values given by reciprocal sums of polynomials.

The meromorphic continuation of the zeta function of Siegel modular threefolds over totally real fields

Thu, 20 Dec 2012 09:16 EST

Cristian Virdol

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 143--148.

Abstract:
In this paper we prove the meromorphic continuation of the zeta function of Siegel modular threefolds over arbitrary totally real number fields.

A relation between the Brauer group and the Tate-Shafarevich group

Thu, 20 Dec 2012 09:16 EST

Chuangxun Cheng

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 149--156.

Abstract:
In this paper, we prove a relation between the Brauer group and the Tate-Shafarevich group for genus one curves over number fields. This is a generalization of a result of Milne in genus one curves case.

Decomposition theorems for Hilbert modular forms

Thu, 20 Dec 2012 09:16 EST

Benjamin Linowitz

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 157--172.

Abstract:
Let $\mathscr{S}_k^+(\mathcal{N},\Phi)$ denote the space generated by Hilbert modular newforms (over a fixed totally real field $K$) of weight $k$, level $\mathcal{N}$ and Hecke character $\Phi$. In this paper we examine the behavior of $\mathscr{S}_k^+(\mathcal{N},\Phi)$ under twists (by a Hecke character). We show how this space may be decomposed into a~direct sum of twists of other spaces of newforms. This sheds light on the behavior of a newform under a~character twist: the exact level of the twist of a newform, when such a~twist is itself a newform, and when a~newform may be realized as the twist of a primitive newform. In certain cases it is shown that the entire space $\mathscr{S}_k^+(\mathcal{N},\Phi)$ can be represented as a direct sum of twists of primitive nebenspaces. This adds perspective to the Jacquet-Langlands correspondence, which characterizes those elements of $\mathscr{S}_k^+(\mathcal{N},\Phi)$ not representable as theta series arising from a quaternion algebra as being precisely those forms which are twists of primitive nebenforms. It follows that in these cases no newforms arise from a quaternion algebra. These results were proven for elliptic modular forms by Hijikata, Pizer and Shemanske by employing the Eichler-Selberg trace formula.

Quadratic residues and class numbers

Thu, 20 Dec 2012 09:16 EST

Wolfgang Knapp, Markus Köcher, Peter Schmid

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 173--182.

Abstract:
For an odd prime $p$ let $\rho_p$ be the least odd prime ($\ne p$) which is a~quadratic residue mod $p$. Using the theorems of Heegner--Baker--Stark and Siegel--Tatuzawa on the class number $h=h(-p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-p})$ it is shown that $\rho_p<\sqrt p$ unless $p\in \{3, 5, 7, 17, 19, 43, 67, 163\}$, possibly with one further exceptional (large) prime $p=p_u$ (satisfying $p=2^{h+2}-u^2$ with $h>100$ und $5\le u<2^{(h-5)/2}$). The exceptional prime does not exist if the Extended Riemann Hypothesis is true.

Diophantine approximation in $\mathbf{Q}(\sqrt{-30})$, $\mathbf{Q}(\sqrt{-33})$\newline and $\mathbf{Q}(\sqrt{-57})$

Thu, 20 Dec 2012 09:16 EST

L. Ya. Vulakh

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 183--205.

Abstract:
For the imaginary quadratic fields with discriminants -120, -132 and -248, the first three, five and two points of the Lagrange and Markov spectra respectively are found.

Circular words and three applications: factors of the Fibonacci word, $\mathcal F$-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,\ldots

Thu, 20 Dec 2012 09:16 EST

Benoît Rittaud, Laurent Vivier

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 207--231.

Abstract:
We introduce the notion of {\em circular words} with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we establish a new result on factors of the Fibonacci word $abaababaabaab\ldots$. Second, we present an expression of the sequence A004146 of [Sloane] in terms of a product of expressions involving roots of unity. Third, we consider the equivalent of $p$-adic numbers that arise by the use of the numeration system defined by the Fibonacci sequence instead of the usual numeration system in base $p$. Among such {\em ${\mathcal F}$-adic numbers}, we give a~characterization of the subset of those which are {\em rational} (that is: a root of an equation of the form $qX=p$, for integral values of $p$ and $q$) by a periodicity property. Eventually, with the help of circular words, we give a complete description of the set of roots of $qX=p$, showing in particular that it contains exactly $q$ ${\mathcal F}$-adic elements.

The sum of digits of polynomial values in arithmetic progressions

Thu, 20 Dec 2012 09:16 EST

Thomas Stoll

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 233--239.

Abstract:
Let $q, m\geq 2$ be integers with $(m,q-1)=1$. Denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Further let $p(x)\in \mathbb{Z}[x]$ be a polynomial of degree $h\geq 3$ with $p(\mathbb{N})\subset \mathbb{N}$. We show that there exist $C=C(q,m,p)>0$ and $N_0=N_0(q,m,p)\geq 1$, such that for all $g\in\mathbb{Z}$ and all $N\geq N_0$, $$\#\{0\leq n< N: s_q(p(n))\equiv g \bmod m\}\geq C N^{4/(3h+1)}.$$ This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is $C N^{2/h!}$.

Numbers with integer expansion in the numeration system with negative base

Thu, 20 Dec 2012 09:16 EST

Petr Ambrož, Daniel Dombek, Zuzana Masáková, Edita Pelantová

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 241--266.

Abstract:
In this paper, we study representations of real numbers in the positional numeration system with negative base, as introduced by Ito and Sadahiro. We focus on the set $\mathbb{Z}_{-\beta}$ of numbers whose representation uses only non-negative powers of $-\beta$, the so-called $(-\beta)$-integers. We describe the distances between consecutive elements of $\mathbb{Z}_{-\beta}$. In case that this set is non-trivial we associate to $\beta$ an infinite word $\boldsymbol{v}_{-\beta}$ over an (in general infinite) alphabet. The self-similarity of $\mathbb{Z}_{-\beta}$, i.e., the property $-\beta \\mathbb{Z}_{-\beta}\subset \mathbb{Z}_{-\beta}$, allows us to find a~morphism under which $\boldsymbol{v}_{-\beta}$ is invariant. On the example of two cubic irrational bases $\beta$ we demonstrate the difference between Rauzy fractals generated by $(-\beta)$-integers and by $\beta$-integers.

Inhomogeneous quadratic congruences

Thu, 20 Dec 2012 09:16 EST

S. Baier, T.D. Browning

Source: Funct. Approx. Comment. Math., Volume 47, Number 2, 267--286.

Abstract:
For given positive integers $a,b,q$ we investigate the density of solutions $(x,y)\in \mathbb{Z}^2$ to congruences $ax+by^2\equiv 0 \bmod{q}$.

Solution of a rational recursive sequences of order three

Mon, 25 Mar 2013 10:47 EDT

E.M. Elsayed

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 7--17.

Abstract:
We obtain in this paper the solutions of the difference equations $$ x_{n+1}=\dfrac{ax_{n}x_{n-2}}{x_{n-1}(-b+cx_{n}x_{n-2})},\qquad n=0,1,..., $$ where $a$, $b$, $c$ are positive real numbers and the initial conditions are arbitrary positive real numbers.

Formal proofs of degree 5 binary BBP-type formulas

Mon, 25 Mar 2013 10:47 EDT

Kundle Adegoke

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 19--27.

A supercongruence for generalized Domb numbers

Mon, 25 Mar 2013 10:47 EDT

Robert Osburn, Brundaban Sahu

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 29--36.

Abstract:
Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a~conjectural supercongruence for numbers which are coefficients in one of Zagier's seven ``sporadic'' solutions to second order Apéry-like differential equations.

On van der Corput property of shifted primes

Mon, 25 Mar 2013 10:47 EDT

Siniša Slijepčević

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 37--50.

Abstract:
We prove that the upper bound for the van der Corput property of the set of shifted primes is $O((\log n)^{-1+o(1)})$, giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes $p-1$. We construct normed non-negative valued cosine polynomials with the spectrum in the set $p-1$, $p\leq n$, and a small free coefficient $a_{0}=O((\log n)^{-1+o(1)})$. This implies the same bound for the intersective property of the set $p-1$, and also bounds for several properties related to uniform distribution of related sets.

Fixed point in CAT(0) spaces

Mon, 25 Mar 2013 10:47 EDT

Ismat Beg, Mujahid Abbas

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 51--59.

Abstract:
We obtained sufficient conditions for existence of fixed points of involutions in CAT$(0)$ spaces. Convergence results of Mann and Ishikawa iterates of weakly contractive mappings are also proved.

Partial sums of the Möbius function in arithmetic progressions assuming GRH

Mon, 25 Mar 2013 10:47 EDT

Karin Halupczok, Benjamin Suger

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 61--90.

Abstract:
We consider Mertens' function in arithmetic progression, \[ M(x,q,a) := \sum{n\le x, n\equiv a mod q} \mu(n). \] Assuming the generalized Riemann hypothesis (GRH), we show that the bound \[ M(x,q,a)\ll_{\varepsilon} \sqrt{x}\exp{((\log x)^{3/5}(\log\log x)^{16/5 +\varepsilon})} \] holds uniform for all $q\le \exp(\frac{\log 2}{2}\lfloor (\log x)^{3/5}(\log\log x)^{11/5}\rfloor)$, $\gcd(a,q)=1$ and all $\varepsilon>0$. The implicit constant is depending only on $\varepsilon$. For the proof, a former method of K. Soundararajan is extended to $L$-series.

Variations of the Ramanujan polynomials and remarks on $\zeta(2j+1)/\pi^{2j+1}$

Mon, 25 Mar 2013 10:47 EDT

Matilde N. Lalín, Mathew D. Rogers

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 91--111.

Abstract:
We observe that five polynomial families have all of their roots on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang [MSW]. Our proofs rely upon theorems of Schinzel [S], and Lakatos and Losonczi [LL] and some generalizations.

Concerning dense subideals in commutative Banach algebras

Mon, 25 Mar 2013 10:47 EDT

Wiesław Żelazko

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 113--115.

Abstract:
In the paper [2] we have shown that in the case of a separable Banach algebra $A$ the necessary and sufficient condition in order that a closed ideal $I\subset A$ has a dense subideal is that $I$ is not finitely (algebraically) generated. We conjectured that this result is true in the general case. In this paper we give an example showing that this conjecture fails to be true.

A maximally separated sequence

Mon, 25 Mar 2013 10:47 EDT

John Bentin

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 117--122.

Abstract:
The paper builds on earlier published work by the author in which a measure for the slowness of clustering of a bounded real sequence, called \emph{separation}, was introduced. Here a conjecture of the earlier paper is proved: that a particular sequence of rational numbers -- the $\mathrm{f}$ sequence -- defined in that paper is of maximal separation.

A remark on the Goldbach-Vinogradov theorem

Mon, 25 Mar 2013 10:47 EDT

Yingchun Cai

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 123--131.

Abstract:
Let $N$ denote a sufficiently large odd integer. In this paper it is proved that $N$ can be represented as the sum of three primes, one of which is $\leq N^{\frac{11}{400}+\varepsilon}$ for any $\varepsilon>0$. This result constitutes an improvement upon that of K.C. Wong, who obtained the exponent $\frac{7}{216}$.

Near-primitive roots

Mon, 25 Mar 2013 10:47 EDT

Pieter Moree

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 133--145.

Abstract:
Given an integer $t\ge 1$, a rational number $g$ and a prime $p\equiv 1 (mod t)$ we say that $g$ is a near-primitive root of index $t$ if $\nu_p(g)=0$, and $g$ is of order $(p-1)/t$ modulo $p$. In the case $g$ is not minus a square we compute the density, under the Generalized Riemann Hypothesis (GRH), of such primes explicitly in the form $\rho(g)A$, with $\rho(g)$ a rational number and $A$ the Artin constant. We follow in this the approach of Wagstaff, who had dealt earlier with the case where $g$ is not minus a square. The outcome is in complete agreement with the recent determination of the density using a very different, much more algebraic, approach due to Hendrik Lenstra, the author and Peter Stevenhagen.

Subfield value sets of polynomials over finite fields

Mon, 25 Mar 2013 10:47 EDT

Wun-Seng Chou, Javier Gomez-Calderon, Gary L. Mullen, Daniel Panario, David Thomson

Source: Funct. Approx. Comment. Math., Volume 48, Number 1, 147--165.

Abstract:
Let $\mathbb{F}_{q^e}$ be a finite field, and let $\mathbb{F}_{q^d}$ be a subfield of $\mathbb{F}_{q^e}$. The \emph{value set} of a polynomial $f$ lying within $\mathbb{F}_{q^d}$ is defined as the set of images $\{f(c) \in \\mathbb{F}_{q^d}\colon c \in \mathbb{F}_{q^e}\}$. This work is concerned with the cardinality of value sets of polynomials lying within subfields.