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arXiv · 1810.10426

A note on the zeros of generalized Hurwitz zeta functions

Giamila Zaghloul
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Abstract

Given a function $f(n)$ periodic of period $q\geq 1$ and an irrational number $0<α\leq 1$, Chatterjee and Gun proved that the series $F(s,f,α)=\sum_{n=0}^{\infty}\frac{f(n)}{(n+α)^s}$ has infinitely many zeros for $σ>1$ when $α$ is transcendental and $F(s,f,α)$ has a pole at $s=1$, or when $α$ is algebraic irrational and $c=\frac{\max{f(n)}}{\min{f(n)}}<1.15$. In this note, we prove that the result holds in full generality.

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Giamila Zaghloul. 2018-10-24. A note on the zeros of generalized Hurwitz zeta functions. https://doi.org/10.1016/j.jnt.2018.09.016

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