On zeros of bilateral Hurwitz and periodic zeta and zeta star functions

TL;DR

This paper investigates the zeros of various bilateral Hurwitz and periodic zeta functions, establishing their locations, simplicity, and conditions for real zeros, along with asymptotic behaviors and complex zeros for rational or transcendental parameters.

Contribution

It provides new results on the distribution, simplicity, and exact locations of zeros of bilateral Hurwitz and periodic zeta functions, including conditions for real zeros and their asymptotic behavior.

Findings

Real zeros of ${ m{Li}}_s (e^{2 heta i})$ do not occur on the real line.

All real zeros of $Y(s,a)$, $O(s,a)$, and $X(s,a)$ are simple and at negative odd integers.

Zeros of $Z(s,a)$ and $P(s,a)$ are simple and at specific integers depending on $a$, with detailed conditions.

Abstract

In this paper, we show the following; (1) The periodic zeta function ${\rm{Li}}_s (e^{2\pi ia})$ with $0<a<1/2$ or $1/2 < a <1$ does not vanish on the real line. (2) All real zeros of $Y(s,a):=\zeta (s,a) - \zeta (s,1-a)$, $O(s,a) := -i {\rm{Li}}_s (e^{2\pi ia}) + i{\rm{Li}}_s (e^{2\pi i(1-a)})$ and $X(s,a) := Y(s,a) + O(s,a)$ with $0 < a < 1/2$ are simple and only at the negative odd integers. (3) All real zeros of $Z(s,a):=\zeta (s,a) + \zeta (s,1-a)$ are simple and only at the non-positive even integers if and only if $1/4 \le a \le 1/2$. (4) All real zeros of $P(s,a):={\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s (e^{2\pi i(1-a)})$ are simple and only at the negative even integers if and only if $1/4 \le a \le 1/2$. Moreover, the asymptotic behavior of real zeros of $Z(s,a)$ and $P(s,a)$ are studied when $0 < a < 1/4$. In addition, the complex zeros of these zeta functions are also discussed when $0 <a <1/2$ is rational or transcendental.