Some Additions to a Family of Sums and Integrals related to Hurwitz' Zeta Function(s), Euler polynomials and Euler Numbers

Abstract

Integrals involving the kernel function $sech(\pi x)$ over a semi-infinite range are of general interest in the study of Riemann's function $\zeta (s)$ and Hurwitz' function $\zeta (s,a)$. Such integrals that include the $arctan$ and $log$ functions in the integrand are evaluated here in terms of $\zeta (s,a)$, thereby adding some new members to a known family of related integrals. A claimed connection between $\zeta (s)$ of odd integer argument and such integrals is verified.

arXiv commentChanged title; improved notation; added new section 3.1.2