Related papers: An identity concerning the Riemann-zeta function

This article provides a proof of the famous \textit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second \textit{Chebyshev Function} $\psi(x)$. We shall be extensively using complex analytic…

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $\operatorname{Im}\log\zeta(1/2+it)$. This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term.…

Let $L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s}$ be a Dirichlet series were $a(n)$ is a bounded completely multiplicative function. We prove that if $L(s)$ extends to a holomorphic function on the open half space $\Re s >1-\delta$,…

A scaling and renormalization approach to the Riemann zeta function, $\zeta$, evaluated at $-1$ is presented in two ways. In the first, one takes the difference between $U_{n}:=\sum_{q=1}^{n}q$ and $4U_{\left\lfloor \frac{n}{2}\right\rfloor…

In this article, we develop a formula for an inverse Riemann zeta function such that for $w=\zeta(s)$ we have $s=\zeta^{-1}(w)$ for real and complex domains $s$ and $w$. The presented work is based on extending the analytical recurrence…

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…

For the multiple zeta function zeta2(s1,s2) of two variables,we obtain its integral representation(involving product of Hurwitz zeta functions) over the interval [1,infinity),with respect to second variable of Hurwitz zeta function and also…

A new method for continuing the usual Dirichlet series that defines the Riemann zeta function ${\zeta}(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.

In this paper, some new results are reported for the study of Riemann zeta function $\zeta(s)$ in the critical strip $0<Re(s)<1$, such as $\zeta(s)$ expressed in a generalized Euler product only involving prime numbers. Particularly, some…

This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is…

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…

Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and Bailey-Borwein-Bradley's identities for generating functions of the sequences $\{\zeta(2n+2)\}_{n\ge 0}, \{\zeta(2n+3)\}_{n\ge 0}.$ By the same method we give…

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = \sigma + it$) is used to obtain estimates for $$…

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…

We investigate the extreme values of the Riemann zeta function $\zeta(s)$. On the 1-line, we obtain a lower bound evaluation $$\max_{t\in[1,T]}|\zeta(1+\i t)|\ge {\rm e}^\gamma(\log_2T+\log_3T+c),$$ with an effective constant $c$ which…

We show for even positive integers $n$ that the quotient of the Riemann zeta values $\zeta(n+1)$ and $\zeta(n)$ satisfies the equation $$\frac{\zeta(n+1)}{\zeta(n)} = (1-\frac{1}{n}) (1-\frac{1}{2^{n+1}-1})…

In this paper we investigate analytic inequalities related to a conjecture of Henry involving the difference between the Riemann zeta function and the digamma function. By treating $\zeta(s)-\psi(1-s)$ as a unified analytic object, we…

In this paper we consider certain 1-parametric family of Dirichlet series. For a particular value of the parameter the series turns into the Dirichlet series for the Riemann zeta function. We prove that almost every series of the family has…

In his second notebook, Ramanujan discovered the following identity for the special values of $\zeta(s)$ at the odd positive integers \begin{equation*}\begin{aligned}\alpha^{-m}\,\left\{\dfrac{1}{2}\,\zeta(2m + 1) + \sum_{n =…

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…