Special Values for the Riemann Zeta Function

The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression for representing the zeta function at the odd integer values 2n+1 for n a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers E2n and the polygamma functions ψ(2n)(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros

The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.

Fast Converging Series for Riemann Zeta Function

Riemann zeta function has a key role in number theory and in its applications. In this paper we present a new fast converging series for . Applications of the series include the computation of the and recursive computation of , and generally . We discuss on the production of irrational number sequences e.g. for encryption coding and zeta function maps for analysis and synthesis of log-time sampled signals.

A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.

Accelerated Series for Riemann Zeta Function at Odd Integer Arguments

Riemann zeta function is an important tool in signal analysis and number theory. Applications of the zeta function include e.g. the generation of irrational and prime numbers. In this work we present a new accelerated series for Riemann zeta function. As an application we describe the recursive algorithm for computation of the zeta function at odd integer arguments.

On the Absence of Zeros of Riemann Zeta-Function Out of ℜ(z) = 1/2

This work shows, after a brief introduction to Riemann zeta function , the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”,, the one Hardy demonstrated in his famous work that infinite countable zeros of the above function can be found on it. Thus, out of this strip, the only remaining zeros of this function are the so-called “trivial ones” . After an analytical introduction reminding the existence of a germ from a generic zero lying in , we show through a Weierstrass-Hadamard representation approach of the above germ that non-trivial zeros out of cannot be found.

Two Theorems on the Zero Density of the Riemann Zeta Function

§1.IntroductionLet N(σ,T) be the number of zeros of the Riemann zeta funetion ζ(s) in theregion σ≤Re(s)≤1,|Im(s)|≤T.For σ>(3/4),by using the Halász-Montgomerymethod,one can get somo results which are better than the classical result givenby Ingham.For example,Jutila proved that the zero density hypothesisN(σ,T)T2-3σ+

On two recurrence formulas for two kinds of identities of Riemann Zeta function

For any complex s, let ζ（s） denote Riemann Zeta function. We have ζ（s）=sum from n=1 to ∞ （1/ns） when Re（s）】1. Now we define A（n,k,l）=sum from α1+α2+……+αk=n to （（α1α2…αk）1ζ（2α1）ζ（2α2）…ζ（2αk））, where n≥k is a positive integer, α+α2+…αk=n denotes the summation for k-dimensional group of positive integers （α1, α2,…, αk）which satisfies this formula. In this note, our main purpose is to discuss computing problem of summation on equation （1）.

Transmission Eigenvalues and the Riemann Zeta Function in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type Ⅰ

We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of Type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles,in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups.For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.

一类新的包含Genocchi数与Riemann Zeta函数求和的计算公式

利用第二类Stirling数,建立了一类含有Genocchi数与Riemann Zeta函数求和的一般计算公式,推广了已有的结果,改进了有关结论.

一个联系Riemann Zeta函数的Hilbert型积分不等式

通过引入独立参量,应用实分析技巧及权函数方法,建立一个最佳常数因子联系Riemann zeta函数的核为余割函数的Hilbert型积分不等式,并导出了其等价式与特殊参数下的齐次形式.

Riemann zeta函数的收敛区域

给出了Riemann zeta函数收敛区域的几种证明.

关于Smarandache函数与Riemann zeta-函数

对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!.即S(n)=min{m:m∈N,n|m!).本文的主要目的是利用初等方法研究一类包含S(n)的Dirichlet级数与Riemann zeta-函数之间的关系,并得到了一个有趣的恒等式.

Riemann Zeta函数ζ(s)的一种推导方法和证明

首先应用三角函数、双曲函数以及二者乘积的级数展开式,证明Riemann Zeta函数ζ(s)(s为偶数)时的一系列表达式,并得到一个表达形式较为简单的递推公式;同时应用此方法得到∑∞n=1coth(πz)n4p+3(p为正整数)时的一个递推公式,并应用留数基本定理逐一证明。

Riemann Zeta函数ξ(2t)(t为正整数)的一个递归公式

本文利用Fourier级数理论得到Riemann Zela函数ξ(s)在s为偶数时的一个递归公式。

一类包含Riemann Zeta函数的求和公式

本文给出了一类包含RiemannZeta函数的求和计算公式 .

一类包含Riemann Zeta函数的求和公式及其估值

本文给出了一类包含RiemannZeta函数的求和计算公式及其估值。

一类包含Riemann Zeta函数的求和公式

应用广义Dirichlet积分公式得到了一类包含RiemannZeta函数的求和计算公式.

Riemann Zeta函数ζ(2n+1)的2个新的表达式

主要研究了ζ函数的表示形式,通过初等及解析的研究方法,给出了关于Riemann Zeta函数ζ(2n+1)的2个新的表达式.

一些与Riemann Zeta函数有关的级数的求和公式

采用组合数学的方法，利用第二类Ｓｔｉｒｌｉｎｇ数和Ｂｅｒｎｏｕｌｉ数给出级数∑∞ｋ＝２ｋｍζ（ｋ）、∑∞ｋ＝１ｋｍζ（２ｋ）及∑∞ｋ＝１（２ｋ＋１）ｍζ（２ｋ＋１）（其中ｍ≥１，ζ（ｘ）＝ζ（ｘ）－１）的求和公式。这些公式表述简洁并有鲜明的规律性。