On some integrals involving the Hurwitz-type Euler zeta functions

Su Hu, Daeyeoul Kim, Min-Soo Kim

Published 2015-08-17Version 1

The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} \zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In algebraic number theory, it represents a partial zeta function of cyclotomic fields in one version of Stark's conjectures (see [12, p.4249, (6.13)]). Its special case, the alternative series \begin{equation*}~\label{Riemann}\zeta_{E}(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}},\end{equation*} has also appeared as a particular situation of the well-known Witten zeta functions in mathematical physics (see [15, p.248, (3.14)]). In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $\zeta_E(s,x)$. There are the analogues of Espinosa and Moll's formulas in [7] for Hurwitz zeta functions.