Infinite products of absolute zeta functions

Fibonacci-Zeta infinite series associated with the polygamma functions

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

Both real and imaginary parts of the function F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s)

The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s).

Infinite matrix products and hypergeometric zeta series

An unpublished identity of Gosper restates a hypergeometric identity for odd zeta values in terms of an infinite product of matrices. We show that this correspondence runs much deeper, and show that many examples of WZ-accelerated series for zeta values lift to infinite matrix products. We also introduce a new matrix subgroup, the Gosper group, which all of our matrix products fall into.

无穷多重ζ星值的算术和与积

Arithmetic sums and products of infinite multiple zeta-star values

本文研究了受限索引的无穷多重ζ星值的算术和与积，并受连分数理论和康托集启发，提出关于特定索引的ζ星值代数点及算术运算的猜想。

黎曼函数Zeta给出了一个对整体损伤的无限的非理性值

La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs

我们提供了一个下界的维数的向量空间由1和黎曼ζ函数的值在第一个奇数。因此，Zeta函数在奇数处取无穷多个无理值。

Riemann Zeta Function. Chi(s)via analytic Continuation inside an infinite Loop

The paper deals with the analytic entire function Chi(s) closely related to Riemann Zeta Function Zeta(s). A formula is obtained for Chi(s) essentially within the so-called critical strip. This is achieved by applying Cauchy integral formula to an infinite loop encircling the critical strip. In the obtained formula a remarkable role is played by a special type of complex function known as Incomplete Gamma Function. Numerical examples verifying the obtained formula are included.

Non-real poles on the axis of absolute convergence of the zeta functions associated to Pascal's triangle modulo a prime

Picking binomial coefficients which cannot be divided by a given prime from Pascal's triangle, we find that they form a set with self-similarity. Essouabri studied on a class of meromorphic functions associated to the above set. These functions are related to fractal geometry and it is a problem whether such a function has a non-real pole on its axis of absolute convergence. Essouabri gave a proof of existence of such a non-real pole in the simplest case. The keys of his proof are Stein's and Wilson's estimates on how fast the points multiply in Pascal's triangle modulo a prime. This article w…

量子细胞自动机zeta函数的绝对zeta函数

Absolute zeta functions for zeta functions of quantum cellular automata

本文证明了量子细胞自动机（QCA）给出的新zeta函数是权取决于构型空间大小的绝对自守形式，并以张量型QCA为例，用多重gamma函数表示其绝对zeta函数，进而通过多重sine函数得到其函数方程。

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned a…

Grover算法与绝对zeta函数

Grover algorithm and absolute zeta functions

本研究揭示了Grover算法与绝对zeta函数之间的联系，通过分析Grover算法的周期，利用Kurokawa定理在有限周期情形下显式构造绝对zeta函数，并给出其展开式。

Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta distributions

In this paper, we give Hurwitz-Lerch zeta distributions with $0 < σ\ne 1$ by using the Gamma function. Moreover, we define Hurwitz-Lerch type of Euler-Zagier double zeta distributions not only in the region of absolute convergence but also the outside of the region of absolute convergence.

Universality of the Hurwitz zeta-function on the half plane of absolute convergence

Let $K$ be a compact set with connected complement on the half-plane Re$(s)>0$, and let $f$ be a continuous function on $K$ which is analytic in its interior. We prove that for any parameter $0<α<1, α\neq \frac 1 2$ then $f(s)$ may be uniformly approximated arbitrarily closely by $ζ(1+iT+iδs,α)$ on $K$ for some $T,δ>0$, where $ζ(s,α)$ denote the Hurwitz zeta-function. This is the first known universality result that is also known to hold for the Hurwitz zeta-function with an algebraic irrational parameter.

On the zeros of generalized Hurwitz zeta functions

In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function.

Functional equations for Selberg zeta functions with Tate motives

For a compact Riemann surface $M$ of genus $g\ge 2$, we study the functional equations of the Selberg zeta functions attached with the Tate motives $f$. We prove that certain functional equations hold if and only if $f$ has the absolute automorphy.

关于绝对导子和Zeta函数的注记

A Note on Absolute Derivations and Zeta Functions

这一评论回答了黑川，落合和和歌山提出的一个问题，是否有一个特定的运营商使用的概念，量子非交换素数的特征值有关的黎曼zeta零点。

低亏格与无限幂零群的Zeta函数结合曲线

Associating curves of low genus to infinite nilpotent groups via the zeta function

从du Sautoy和Grunewald在\cite{duSG1}中的工作可知，计算无限幂零群中的有限指数子群的zeta函数依赖于代数簇的某些相关系统在约化$\modp.$上的行为。在\cite{duS3, duS4} du Sautoy中构造了一个群，其局部zeta函数由椭圆曲线$E:Y^2=X^3-X.$上的点的数目确定。的构造来定义一类群，其局部zeta函数依赖于具有有理点的给定椭圆曲线的约化上的点数。我们还构造了一类群，其行为与亏格2的任何曲线具有有理点的行为相同。最后我们讨论了这项工作中出现的问题。

On gaps between zeros of the Riemann zeta function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.

关于量子游走与绝对 zeta 函数之间的关系

On the relation between quantum walks and absolute zeta functions

本文建立图上 Grover walk 与绝对 zeta 函数的联系，证明量子游走诱导的 zeta 函数构成依赖边数的绝对自守形式，并以循环图为例将其绝对 zeta 函数表示为二阶多重 gamma 函数。

On the Distribution of the non-trivial Roots of Riemann's Zeta Function. An alternative Approach

Expressing Weierstrass type infinite products in terms of Stieltjes integrals is discussed. The asymptotic behavior of particular types of infinite products is compared against the asymptotic behavior of the entire function Xi(s), well-known in Riemann zeta function theory. An approximate formula for the distribution of the non-trivial roots of Riemann's zeta function is obtained.

A note on the gaps between consecutive zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.5155 times the average spacing and infinitely often they differ by at least 2.69 times the average spacing.

Ihara Zeta functions of infinite weighted graphs

The theory of Ihara zeta functions is extended to infinite graphs which are weighted and of finite total weight. In this case one gets meromorphic instead of rational functions and the classical determinant formulas of Bass and Ihara hold true with Fredholm determinants.

非伽罗瓦三次域的zeta函数的中心值

Central values of zeta functions of non-Galois cubic fields

该研究证明了无穷多个非伽罗瓦三次域的戴德金zeta函数中心值为负，揭示了这类数域zeta函数在中心点的符号性质。

The Ihara zeta function of the infinite grid

The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite. The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta f…

Hurwitz Zeta Functions and Ramanujan's Identity for Odd Zeta Values

Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing the integral kernel associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.

通过锥形zeta值的Dedekind和互反律

Reciprocity For Dedekind Sums via Conical Zeta Values

研究绝对连续函数相关Dedekind和的互反公式，推广经典Dedekind-Rademacher互反律，利用Fourier分析揭示多项式型函数互反性的锥形zeta值几何解释。

动态zeta函数与Kummer同余

Dynamical zeta functions and Kummer congruences

我们建立了Artin-Mazur zeta-函数的系数和Kummer同余之间的联系。这允许肯定地解决存在映射T的问题，使得T的第n阶不动点的数目等于第2n阶欧拉数的绝对值。并解决了一个与Riemann-Siegel公式系数有关的Gabcke问题。

Large gaps between consecutive zeros of the Riemann zeta-function. II

Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.

On large gaps between zeros of the Riemann zeta-function

Assuming the Generalized Riemann Hypothesis(GRH), we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 3.072 times the average spacing.

The summation of infinite partial fraction decomposition I: some formulae related to the Hurwitz zeta function

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.

代数数高度的下界与 Dedekind zeta 函数的低位零点

Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function

本文借助 Dedekind zeta 函数低位零点建立代数整数 Weil 高度下界，并在 GRH 条件下证明特定无限非 Galois 扩张中的 Lehmer 猜想。

Hurwitz zeta and Euler-Zagier-Hurwitz type of double zeta distributions and real zeros of these zeta functions

In this paper, we give Hurwitz zeta distributions with $0 < σ\ne 1$ by using the Gamma function. During the proof process, we show that the Hurwitz zeta function $ζ(σ,a)$ does not vanish for all $0 <σ<1$ if and only if $a \ge 1/2$. Next we define Euler-Zagier-Hurwitz type of double zeta distributions not only in the region of absolute convergence but also the outside of the region of absolute convergence. Moreover, we show that the Euler-Zagier-Hurwitz type of double zeta function $ζ_2 (σ_1,σ_2\,;a)$ does not vanish when $0<σ_1<1$, $σ_2>1$ and $1<σ_1+σ_2<2$ if and only if $a \ge 1/2$.

多重zeta值、WZ对与无穷和计算

Multiple zeta values, WZ-pairs and infinite sums computations

本研究结合Wilf-Zeilberger对方法与多重zeta值系统理论，证明了Z.W. Sun提出的大量级数恒等式，破解了诸多长期悬而未决的猜想。

The divisibility of zeta functions of cyclotomic function fields

In this paper, we generalize Bernoulli-Goss polynomials, and give a criterion on the divisibility of zeta functions of cyclotomic function fields. As an application of our criterion, for a given polynomial $f(u)$, we prove that there are infinitely many cyclotomic function fields whose zeta polynomial is divided by $f(u)$.

On the Montgomery-Odlyzko method regarding gaps between zeros of the zeta-function

Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.

黎曼ζ函数零点之间的大间隙

Large gaps between the zeros of the Riemann zeta function

证明了广义黎曼假设意味着黎曼ζ函数存在无穷多个连续零点，其零点间距是平均零点间距的2.9125倍.这是从计算黎曼ζ函数的二阶矩乘以在ζ函数的零点上平均的狄利克雷多项式推导出来的。

Values of zeta-one functions at positive even integers

Motivated by Euler-Goldbach and Shallit-Zikan theorems, we introduce zeta-one functions with infinite sums of $n^{s}\pm1$ as an analogy of the Riemann zeta function. Then we compute values of these functions at positive even integers by the residue theorem.

特征$p$ zeta函数零点的消失级

Orders of vanishing of zeros of characteristic $p$ zeta function

zeta函数零点的消失级包含了大量的算术信息。对于绝对zeta函数，当有限域有两个元素时，Dinesh Thakur给出了其零点阶为零的充分条件.这样的条件只考虑主理想。这一结果由Thakur和Diaz-Vargas推广。现在，条件不仅涉及主理想，而且还涉及理想的所有类，仍然在两个元素的域中。在这项工作中，我们使用Thakur和Diaz-Vargas的类似证明，将这些结果推广到任意有限域。

A note on some infinite sums of Hurwitz zeta functions

We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $ζ(s,α)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $ζ(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.

Asymptotic expansions for partitions generated by infinite products

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $\mathcal{L}$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree $n$ of $\mathfrak{so}{(5)}$. We also study the Witten zeta function $ζ_{\mathfrak{so…

Opposite Sign Kloosterman Sum Zeta Function

We study the meromorphic continuation and the spectral expansion of the oppposite sign Kloosterman sum zeta function, $$(2π\sqrt{mn})^{2s-1}\sum_{\ell=1}^\infty \frac{S(m,-n,\ell)}{\ell^{2s}}$$ for $m,n$ positive integers, to all $s \in \mathbb{C}$. There are poles of the function corresponding to zeros of the Riemann zeta function and the spectral parameters of Maass forms. The analytic properties of this function are rather delicate. It turns out that the spectral expansion of the zeta function converges only in a left half-plane, disjoint from the region of absolute convergence of the Diric…

A complement to a recent paper on some infinite sums with the zeta values

Recently, several new results related to the evaluation of the series sum (-1)^n zeta(n)/(n+k) were published. In this short note we show that this series also possesses an interesting connection to the values of the zeta-function on the critical line and to the Euler constant.

Simple functional equations for generalized Selberg zeta functions with Tate motives

We prove that for a compact locally symmetric Riemannian space $M$ of rank 1 there exist infinitely many automorphic Tate motives $f$ such that the generalized Selberg zeta function $Z_{M(f)}(s)$ satisfies a simple functional equation in the sense that it has no gamma factors.

Gaps between zeros of the Riemann zeta-function

We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).

Zeros of partial sums of the Dedekind zeta function of a cyclotomic field

In this article, we study the zeros of the partial sums of the Dedekind zeta function of a cyclotomic field $K$ defined by the truncated Dirichlet series \[ ζ_{K, X} (s) = \sum_{\|\mathfrak{a}\| \leq X} \frac{1}{\|\mathfrak{a}\|^{s}}, \] where the sum is to be taken over nonzero integral ideals $\mathfrak{a}$ of $K$ and $\|\mathfrak{a}\|$ denotes the absolute norm of $\mathfrak{a}$. Specifically, we establish the zero-free regions for $ζ_{K, X} (s)$ and estimate the number of zeros of $ζ_{K, X} (s)$ up to height $T$.

Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function

The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An analytic continuation formula for these hypergeometric functions exists and is used to derive some infinite sums which allow the zeta function at integer arguments n to be written as a weighted infinite sum of hypergeometric functions at n - 1. The form might be considered to be a shift operator for the Riemann zeta function which leads to the curious values ζF(0)…

涉及Riemann zeta函数的两个函数的单调性和绝对凸性

Monotonicity and absolute convexity of two functions involving Riemann zeta function

利用Riemann zeta函数的积分表示、单调性规则及函数1/(e^t-1)的性质，研究两个函数的单调性和凸性，并推导出Bernoulli数比值的单调性和不等式。

Computing the residue of the Dedekind zeta function

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33 log D_K/(\sqrt{X}\log X), where D_K is the absolute value of the discriminant of K. Guided by Weil's explicit formula and still assuming GRH, we make a different use of the splitting of primes and thereby improve Bach's constant to 2.33. This results in substantial speeding of one part of Buchmann's class group algorithm.

Horizontal Monotonicity of the Modulus of the Riemann Zeta Function and Related Functions

It is shown that the absolute values of Riemann's zeta function and two related functions strictly decrease when the imaginary part of the argument is fixed to any number with absolute value at least 8 and the real part of the argument is negative and increases up to 0; extending this monotonicity to the increase of the real part up to 1/2 is shown to be equivalent to the Riemann Hypothesis. Another result is a double inequality relating the real parts of the logarithmic derivatives of the three functions under consideration.

关于退化调和数与退化高阶调和数的一些恒等式

Some identities on degenerate harmonic and degenerate higher-order harmonic numbers

本文研究退化调和数与退化高阶调和数，证明其相关无穷和可用退化zeta函数及退化Hurwitz zeta函数表示。

Large gaps between consecutive zeros of the Riemann zeta-function

Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.033 times the average spacing.

多重有限黎曼ζ函数

Multiple finite Riemann zeta functions

观察多个版本的除数函数，我们引入一个新的zeta函数，我们称之为多重有限黎曼ζ函数。利用$q$-级数恒等式证明了zeta函数有一个欧拉积，并描述了zeta函数零点的位置。我们进一步研究多变量和多参数版本的多个有限黎曼ζ函数和他们的无限对应与对称多项式和一些算术量称为强大的数字。

Harmonic-Number Summation Identities, Symmetric Functions, and Multiple Zeta Values

We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently conjectured by J. Choi, and give several more families of identities of a similar nature.

On $r$-gaps between zeros of the Riemann zeta-function

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(γ_{n+r}-γ_n)/(2π/\log γ_n) > r + Θ\sqrt{r}$ and $(γ_{n+r}-γ_n)/(2π/\log γ_n) < r - \vartheta\sqrt{r}$ for explicit absolute positive constants $Θ$ and $\vartheta$, where $γ$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontriv…

On the value-distributions of logarithmic derivatives of Dedekind zeta functions

We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any algebraic number field. We construct the density functions as the Fourier inverse transformations of certain functions represented by infinite products that come from the Euler products of the Dedekind zeta functions.