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

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 ${\int }_{Res=c}\frac{\zeta (s)}{s}ds$ and ${\int }_{Res=c}\frac{\eta (s)}{s}ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form.

17 pages, no figures

References in corpus (4)

• Series representations for the Stieltjes constants
• Sums of alternating products of Riemann zeta values and solution of a Monthly problem
• Evaluation of some second moment and other integrals for the Riemann, Hurwitz, and Lerch zeta functions
• Parameterized summation relations for the Stieltjes constants