On the Hurwitz zeta function of imaginary second argument

Abstract

In this work, we exploit Jonquière's formula relating the Hurwitz zeta function to a linear combination of polylogarithmic functions in order to evaluate the real and imaginary part of ζH(s, ia) and its first derivative with respect to the first argument s. In particular, we obtain expressions for the real and imaginary party of ζH(s, ia) and its derivative for s = m with \documentclass[12pt]{minimal}\begin{document}$m\in \mathbb {Z}\backslash \lbrace 1\rbrace$\end{document}m∈Z∖{1} involving simpler transcendental functions. We apply these results to the computation of the imaginary part of the one-loop effective action for massive scalar fields under the influence of a strong electric field in higher dimensional Minkowski spacetime.