A note on some infinite sums of Hurwitz zeta functions

R B Paris

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arXiv · 2104.00957

A note on some infinite sums of Hurwitz zeta functions

R B Paris

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Abstract

We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $\zeta(s,\alpha)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m \zeta(s,k),\] 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 $a\to0$.

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R B Paris. 2021-04-02. A note on some infinite sums of Hurwitz zeta functions. https://arxiv.org/abs/2104.00957

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