On a Weighted Series of the Hurwitz Zeta Function

TL;DR

This paper proves that certain weighted series involving the Hurwitz zeta function can be expressed as finite combinations of Hurwitz (Lerch) zeta functions, extending the theory of higher-order convex functions.

Contribution

It establishes a new finite representation for weighted series of the Hurwitz zeta function, linking it to higher-order convex function theory.

Findings

Series resolve into finite combinations of Hurwitz (Lerch) zeta functions.

Applicable for all natural numbers a, non-negative real x, and complex s with Re(s) > a + 2.

Extends the Bohr-Mollerup theorem to higher-order convex functions.

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.