Zeros and the functional equation of the quadrilateral zeta function

TL;DR

This paper investigates the zeros of various bilinear and quadrilateral zeta functions, establishing their locations on the real line and analyzing their complex zeros, with implications for the Riemann hypothesis.

Contribution

It characterizes the real zeros of the bilateral Hurwitz, periodic, and quadrilateral zeta functions, extending known zero distributions and exploring their complex zeros and functional equations.

Findings

Real zeros of $Z(s,a)$ are on non-positive even integers.

Real zeros of $P(s,a)$ are on negative even integers.

Real zeros of $Q(s,a)$ are on negative even integers.

Abstract

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 only the non-positive even integers exactly same as in the case of $(2^s-1) \zeta (s)$. 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 only the negative even integers just like $\zeta (s)$. Moreover, we show 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 only the negative even integers. On the other hand, we prove that $Z(s,a)$, $P(s,a)$ and $Q(s,a)$ have at least one real zero in $(0,1)$ when $0<a<1/2$ is sufficiently small. The complex zeros of these zeta functions are also discussed when $1/4 \le a \le 1/2$ is rational or transcendental. As a corollary, we show that $Q(s,a)$ with rational $1/4 < a < 1/3$ or $1/3 < a < 1/2$ does not satisfy the analogue of the Riemann hypothesis even though $Q(s,a)$ satisfies the functional equation that appeared in Hamburger's or Hecke's theorem and all real zeros of $Q(s,a)$ are located at only the negative even integers again as in the case of $\zeta (s)$.