A note on some infinite sums of Hurwitz zeta functions

Abstract

We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $\zeta (s,\alpha )$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $\zeta (s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to 0$.

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