A note on the zeros of generalized Hurwitz zeta functions

TL;DR

This paper extends previous results on the zeros of generalized Hurwitz zeta functions, showing that they have infinitely many zeros under broader conditions, regardless of the nature of the irrational parameter.

Contribution

It generalizes earlier findings by proving the zero distribution of these functions in full generality, removing previous restrictions on the parameter $eta$.

Findings

The series $F(s,f,eta)$ has infinitely many zeros for $ ext{Re}(s)>1$ under broad conditions.

The result applies to all irrational $eta$, including algebraic and transcendental cases.

The paper confirms the zero distribution behavior without restrictions on the nature of $eta$.

Abstract

Given a function $f(n)$ periodic of period $q\geq 1$ and an irrational number $0<\alpha\leq 1$, Chatterjee and Gun proved that the series $F(s,f,\alpha)=\sum_{n=0}^{\infty}\frac{f(n)}{(n+\alpha)^s}$ has infinitely many zeros for $\sigma>1$ when $\alpha$ is transcendental and $F(s,f,\alpha)$ has a pole at $s=1$, or when $\alpha$ 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.