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

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.