On Lerch's formula and zeros of the quadrilateral zeta function

Abstract

Let $0<a\le 1/2$ and define the quadrilateral zeta function by $2Q(s,a):=\zeta (s,a)+\zeta (s,1-a)+{\mathrm{Li}}_s(e^{2\pi ia})+{\mathrm{Li}}_s(e^{2\pi i(1-a)})$, where $\zeta (s,a)$ is the Hurwitz zeta function and ${\mathrm{Li}}_s(e^{2\pi ia})$ is the periodic zeta function. In the present paper, we show that there exists a unique real number $a_0\in (0,1/2)$ such that $Q(\sigma ,a_0)$ has a unique double real zero at $\sigma =1/2$ when $\sigma \in (0,1)$, for any $a\in (a_0,1/2]$, the function $Q(\sigma ,a)$ has no zero in the open interval $\sigma \in (0,1)$ and for any $a\in (0,a_0)$, the function $Q(\sigma ,a)$ has at least two real zeros in $\sigma \in (0,1)$. Moreover, we prove that $Q(s,a)$ has infinitely many complex zeros in the region of absolute convergence and the critical strip when $a\in \mathbb{Q}\cap (0,1/2)\setminus \{1/6,1/4,1/3\}$. The Lerch formula, Hadamard product formula, Riemann-von Mangoldt formula for $Q(s,a)$ are also shown.

arXiv comment16 pages, 10 figures. The title is changed. Theorem 1.5 and Corollary 1.6 are added