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 throughout 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 $\int _{\mbox {\tiny {Re}} s=c} {{\zeta (s)} \over s} ds$ and $\int _{\mbox {\tiny {Re}} s=c} {{\eta (s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form.