On zeros of the bilateral Hurwitz and periodic zeta functions

Takashi Nakamura

Published 2017-12-14Version 1

In this paper, we show that all real zeros of the bilateral Hurwitz zeta function $Z(s,a):=\zeta (s,a) + \zeta (s,1-a)$ with $1/4 \le a \le 1/2$ are on the non-positive even integers as well as the real zeros of $\zeta (s,1/2) = (2^s-1) \zeta (s)$. We also prove that all real zeros of the bilateral periodic zeta function $P(s,a):={\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s (e^{2\pi i(1-a)})$ with $1/4 \le a \le 1/2$ are on the negative even integers as well as the real zeros of $\zeta (s)$. Moreover, we show that that all real zeros of the quadrilateral zeta function $Q(s,a):=Z(s,a) + P(s,a)$ with $1/4 \le a \le 1/2$ are on the non-positive even integers. The complex zeros of these zeta functions are also discussed.