Related papers: On Multiple Zeta Values of Even Arguments

Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|\zeta(1/2+it)|^2 dt - T\Bigl(\log {T\over2\pi} + 2\gamma -1\Bigr)$$ is the error term in the mean square formula for $|\zeta(1/2+it)|$.…

In a recent work, Dancs and He found new `Euler-type' formulas for $\,\ln{2}\,$ and $\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, each containing a series that apparently can not be evaluated in closed form, distinctly from…

In this paper, we investigate three general forms of multiple zeta(-star) values. We use these values to give three new sum formulas for multiple zeta(-star) values with height $\leq 2$ and the evaluation of…

We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it…

For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive…

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…

Let $Z_n(s; a_1,..., a_n)$ be the Epstein zeta function defined as the meromorphic continuation of the function \sum_{k\in\Z^n\setminus\{0\}}(\sum_{i=1}^n [a_i k_i]^2)^{-s}, \text{Re} s>\frac{n}{2} to the complex plane. We show that for…

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure…

A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.

Let $m$ be a positive integer, and define $$\zeta_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\omega(n)}}{n^s}\ \ \ \ \text{and} \ \ \ \ \zeta^*_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\Omega(n)}}{n^s},$$ for $\Re(s)>1$, where…

For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $\zeta(i_1,...,i_k)$ with odd denominators. Like the…

In this article we shall survey some recent progress on the study of Ap\'ery-like sums which are multiple variable generalizations of the two sums Ap\'ery used in his famous proof of the irrationality of $\zeta(2)$ and $\zeta(3)$. We only…

We prove formulas for the Bernoulli numbers by using the Newton-Girard identities to evaluate the Riemann zeta function at positive even integers. To do this, we define a sequence of positive integers, a sequence of polynomials, and a…

For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$…

We present results for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-dimensional and two-dimensional series. Most of these series can be expressed in…

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…

We show that if a,b,c,d,f are positive integers such that a+b+c+d+f is even, then the Witten zeta value zeta_{sl(4)}(a,b,c,d,0,f) is expressible in terms of Witten zeta functions with fewer arguments.

By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $\zeta(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such…

We give thirty-two diverse proofs of a small mathematical gem--the fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many…

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}%…