ON ZEROS OF BILATERAL HURWITZ AND PERIODIC ZETA AND ZETA STAR FUNCTIONS

Abstract

We show that

• (1) the periodic zeta function ${\text{Li}}_s(e^{2\pi ia})$ with $0<a<\frac{1}{2}$ or $\frac{1}{2}<a<1$ does not vanish on the real line;

• (2) all real zeros of $Y(s,a):=\zeta (s,a)-\zeta (s,1-a)$, $O(s,a):=-i{\text{Li}}_s(e^{2\pi ia})+i{\text{Li}}_s(e^{2\pi i(1-a)})$ and $X(s,a):=Y(s,a)+O(s,a)$ with $0<a<\frac{1}{2}$ are simple and are located only at the negative odd integers;

• (3) all real zeros of $Z(s,a):=\zeta (s,a)+\zeta (s,1-a)$ are simple and are located only at the nonpositive even integers if and only if $\frac{1}{4}\le a\le \frac{1}{2}$;

• (4) all real zeros of $P(s,a):={\text{Li}}_s(e^{2\pi ia})+{\text{Li}}_s(e^{2\pi i(1-a)})$ are simple and are located only at the negative even integers if and only if $\frac{1}{4}\le a\le \frac{1}{2}$.

Moreover, the asymptotic behavior of real zeros of $Z(s,a)$ and $P(s,a)$ are studied when $0<a<\frac{1}{4}$. In addition, the complex zeros of these zeta functions are also discussed when $0<a<\frac{1}{2}$ is rational or transcendental.