On the Hurwitz Zeta Function of Imaginary Second Argument

Abstract

In this work we exploit Jonqui\`{e}re's formula relating the Hurwitz zeta function to a linear combination of polylogarithmic functions in order to evaluate the real and imaginary part of $\zeta_{H}(s,ia)$ and its first derivative with respect to the first argument $s$. In particular, we obtain expressions for the real and imaginary party of $\zeta_{H}(s,i a)$ and its derivative for $s=m$ with $m\in\mathbb{Z}\backslash\{1\}$ involving simpler transcendental functions.