An Explicit Formula Relating Stieltjes Constants and Li's Numbers

This study deals with certain harmonic zeta functions, one of them occurs in the study of the multiplication property of the harmonic Hurwitz zeta function. The values at the negative even integers are found and Laurent expansions at poles are described. Closed-form expressions are derived for the Stieltjes constants that occur in Laurent expansions in a neighborhood of s=1. Moreover, as a bonus, it is obtained that the values at the positive odd integers of three harmonic ze...

In this paper, we consider meromorphic extension of the function \[ \zeta_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call \textit{hyperharmonic zeta function}) where $h_{n}^{(r)}$ are the hyperharmonic numbers. We establish certain constants, denoted $\gamma_{h^{\left( r\right) }}\left( m\right) $, which naturally occur in the Laurent expansion of $\zeta_{h^{\left(...

Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the $\Gamma$-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of $\gamma_1(1/2)$, $\gamma_1(1/3)$, $\gamma_1(2/3)$, $\gamma_1(1/4)$, $\gamma_1(3/4)$, $\gamma_1(1/6)$, $\gamma_1(5/6)$, which could be expressed ...

This work is dedicated to the promotion of the results Hadamard, Landau E., Walvis A., Estarmann T and Paul R. Chernoff for pseudo zeta functions. The properties of zeta functions are studied, these properties can lead to new regularities of zeta functions.

The aim of this paper is to present a revised version of my proof of the Riemann Hypothesis in which a few more details and explanations have been added

This review article brings forth some recent results in the theory of the Riemann zeta-function $qzeta(s)$.

The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable f...

In this expository article, we discuss the contributions made by several mathematicians with regard to a famous formula of Ramanujan for odd zeta values. The goal is to complement the excellent survey by Berndt and Straub \cite{berndtstraubzeta} with some of the recent developments that have taken place in the area in the last decade or so.

A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants is derived in this paper. In addition, we also report a formula for the Stieltjes constants in terms of the higher derivatives of the Riemann zeta function. A formula for the Stieltjes constants in terms of the (exponential) complete Bell polynomials containing the eta constants as the arguments is also derived.

In 1730, Euler defined the Gamma function $\Gamma(x)$ by the integral representation. It possesses many interesting properties and has wide applications in various branches of mathematics and sciences. According to Lerch, the Gamma function $\Gamma(x)$ can also be defined by the derivative of the Hurwitz zeta function $$\zeta(z,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^{z}}$$ at $z=0$. Recently, Hu and Kim defined the corresponding Stieltjes constants $\widetilde{\gamma}_{k}(x)$ a...