The functional equation and zeros on the critical line of the quadrilateral zeta function

Takashi Nakamura

Published 2019-10-22Version 1

Let $0 < a \le 1/2$ and define the quadrilateral zeta function by $Q(s,a) := \zeta (s,a) + \zeta (s,1-a) + {\rm{Li}} (s,a) + {\rm{Li}} (s,1-a)$, where $\zeta (s,a)$ is the Hurwitz zeta function and ${\rm{Li}} (s,a)$ is the periodic zeta function. In the present paper, we prove that for any $0 < a \le 1/2$, there are positive constants $A(a)$ and $T_0(a)$ such that the numbers of zeros of $Q(s,a)$ on the line segment from $1/2$ to $1/2 +iT$ is at least $A(a) T$ whenever $T \ge T_0(a)$. We also give some remarks on the functional equations of zeta functions related to Hamburger's and Knopp's theorems.