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arXiv · 2302.01865

On a Weighted Series of the Hurwitz Zeta Function

Matthew Fox·Chaitanya Karamchedu
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Abstract

In this note we prove that for all $a \in \mathbb{N}$, $x \in \mathbb{R}_+ \cup \{0\}$, and $s \in \mathbb{C}$ with $\Re(s) > a + 2$, the (alternating) weighted series of the Hurwitz zeta function, $$ \sum_{k \geq 1} (\pm 1)^k (k + x)^a\zeta(s,k + x), $$ resolves into a finite combination of Hurwitz (Lerch) zeta functions. This applies in Marichal and Zena\"idi's theory on analogues of the Bohr-Mollerup theorem for higher-order convex functions.

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Matthew Fox, Chaitanya Karamchedu. 2023-02-03. On a Weighted Series of the Hurwitz Zeta Function. https://arxiv.org/abs/2302.01865

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