Zeros of the Hurwitz zeta function in the interval (0,1)

TL;DR

This paper establishes conditions under which the Hurwitz zeta function has no zeros in certain intervals, providing new insights into its zero distribution and confirming known results for the classical zeta function.

Contribution

The paper introduces a new zero-free region for the Hurwitz zeta function based on parameter conditions, extending understanding of its zeros.

Findings

Hurwitz zeta function is zero-free in certain parameter regions

Classical zeta function has no zeros in (0,1)

Provides conditions for negativity of the Hurwitz zeta function

Abstract

We first give a condition on the parameters $s,w$ under which the Hurwitz zeta function $\zeta(s,w)$ has no zeros and is actually negative. As a corollary we derive that it is nonzero for $w\geq 1$ and $s\in(0,1)$ and, as a particular instance, the known result that the classical zeta function has no zeros in $(0,1)$.