The Mean-Square of the Lerch Zeta-Function Near the Critical Line

In the paper, the asymptotics of the mean-square of the Lerch zeta-function near the critical line is obtained. Three different definitions of the closeness to the critical line are considered.     

On Functional Relations for the Alternating Analogues of Tornheim's Double Zeta Function

In this paper, new proofs of two functional relations for the alternating analogues of Tornheim's double zeta function are given. Using the functional relations, the author gives new proofs of some evaluation formulas found by Tsumura for these alternating series.     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

Explicit Approximate Functional Equations for Various Classes of Dirichlet Series

We give explicit approximate functional equations for various classes of Dirichlet series, such as Lerch zeta function or Dirichlet L-series.Dedicated to Jean-Louis Nicolas on the occasion of his sixtieth birthday2000 Mathematics Subject Classification: Primary—11M06, 11M35, 11Y60     

The representation of the generalized linear Euler sums with parameters

In this paper, by using residue method, we obtain the representations of some basic linear generalized Euler sums with parameters. Based on the linear generalized Euler sums with parameters, some new Euler sums are obtained and expressed in the closed forms. When the parameters of new Euler sums take special values, we can get some usual expressions of Euler sums. Moreover, the integrals of many special functions can be expressed as the Euler sums given in this paper.     

Zeta functions interpolating the convolution of the Bernoulli polynomials

We prove several relations on multiple Hurwitz–Riemann zeta functions. Using analytic continuation of these multiple Hurwitz–Riemann zeta functions, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers. As an application, we give a general convolution identity for Bernoulli numbers.     

Some Unusual Identities for Special Values of the Riemann Zeta Function

In this paper, we use elementary methods to derive some new identities for special values of the Riemann zeta function.     

On Some Integrals Involving the Hurwitz Zeta Function: Part 2

We establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, ln sin(q), ln (q) and the polygamma functions. Many of the results are most conveniently formulated in terms of a family of functions A _k(q) := k(1 – k, q), k , and a family of polygamma functions of negative order, whose properties we study in some detail.     

On Some Integrals Involving the Hurwitz Zeta Function: Part 1

We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, ln (q) and ln sin(q).     

Hermite Expansion of the Riemann Zeta Function

Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction, it is crucial to regard the zeta function as Fourier transfomations of Schwartz' distributions.     

On Solutions to Riemann's Functional Equation

In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta function in the space of functions representable by ordinary Dirichlet series satisfying certain regularity conditions. We consider solutions to a more general functional equation with real weight k. In the case of Hamburger's theorem, k = . We show that, under suitable conditions, the generalized functional equation admits no nontrivial solutions for k > 0 unless k = . Our proof generalizes an elegant proof of Hamburger's theorem given by Siegel, and employs a generalized integral transform.1997 Sunrise Way     

Riemann Zeta函数的六阶和

利用概率论与组合数学的方法,研究了与Riemann-zeta函数ξ(k)的部分和ξ_n(k)有关的一些级数,计算出了一些重要的和式．特别的,Euler的著名结果5ξ(4)= 2ξ~2(2)能够从四阶和式直接推出．因此,通过计算全部的11个六阶和式,研究它们之间的非平凡关系,就有可能得到ξ(3)的数值．     

Vinogradov's Integral and Bounds for the Riemann Zeta Function

The main result is an upper bound for the Riemann zeta functionin the critical strip: with A = 76.2 and B = 4.45, valid for 1 and |t| 3. The previousbest constant B was 18.5. Tools include a variant of the Korobov–Vinogradovmethod of bounding exponential sums, an explicit version ofT. D. Wooley's bounds for Vinogradov's integral, and explicitbounds for mean values of exponential sums over numbers withoutsmall prime factors, also using methods of Wooley. An auxiliaryresult is the exponential sum bound , where N is a positive integer, t is a real number, = log (t)/(logN) and 2000 Mathematical Subject Classification: primary 11M06, 11N05,11L15; secondary 11D72, 11M35.     

New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

Twists of Lerch Zeta-Functions

We study twists of the Lerch zeta-functions with shifted Dirichlet characters. This generalizes Dirichlet L-functions. These twists turn out to be meromorphic with at most one simple pole and satisfy a functional equation (similar to the functional equation of the Lerch zeta-function) if the Dirichlet character is primitive. We also investigate the distributions of their zeros. In particular, we disprove the analogue of Riemann's hypothesis for some twists.     

Omega Theorems for Zeta Sums

We obtain lower bounds for the moduli of trigonometric sums in the theory of Riemann zeta functions.     

Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions: III

The main object of this paper is the mean square I _h (s) of higher derivatives of Hurwitz zeta functions (s, ). We shall prove asymptotic formulas for I _h (1/2 + it) as t + with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for I _h (1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for I _h (1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product (u, )(v, ) with the independent complex variables u and v.     

Zeta Regularizations

We survey our work on various generalizations of the usual zeta regularized product and their applications. For construction of functions having a certain invariance and a given set of zeros, so-called the zeta regularization method is quite useful. We exhibit several examples and discuss some questions.     

Sums Involving the Hurwitz Zeta Function

We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers.     

一类由函数方程确定的周期函数

通过对一例题的分析,讨论了一类由函数方程确定的周期函数.