Real zeros of Hurwitz zeta-functions and their asymptotic behavior in the interval $(0,1)$

TL;DR

This paper investigates the zeros of Hurwitz zeta-functions within the interval (0,1), establishing the existence and uniqueness of zeros depending on the parameter a, and analyzing their asymptotic behavior.

Contribution

It demonstrates the precise conditions for the existence of a unique zero of the Hurwitz zeta-function in (0,1) and describes its asymptotic behavior as a varies.

Findings

Unique zero exists in (0,1) for 0<a<1/2.

No zeros in (0,1) for 1/2≤a≤1.

Asymptotic behavior of the zero as a approaches 0 or 1/2.

Abstract

Let $0<a\leq1, s\in\mathbb{C}$, and $\zeta(s,a)$ be the Hurwitz zeta-function. Recently, T.~Nakamura showed that $\zeta(\sigma,a)$ does not vanish for any $0<\sigma<1$ if and only if $1/2\leq a \leq1$. In this paper, we show that $\zeta(\sigma,a)$ has precisely one zero in the interval $(0,1)$ if $0<a<1/2$. Moreover, we reveal the asymptotic behavior of this unique zero with respect to $a$.