Series representation of the Riemann zeta function and other results: Complements to a paper of Crandall

TL;DR

This paper introduces new series representations for the Riemann and Hurwitz zeta functions, providing analytic continuation across the complex plane, and explores related series for Stieltjes constants and integral evaluations.

Contribution

It offers alternative series representations for key special functions, extending their analytic properties and computational approaches, complementing prior work by Crandall.

Findings

New series representations for Riemann and Hurwitz zeta functions

Series expressions for Stieltjes constants in hypergeometric form

Evaluations of integrals involving zeta and eta functions

Abstract

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form.