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 all real zeros of Q(s, a) are simple and are located only at the negative even integers just like \(\zeta (s)\) if and only if \(a_0 < a \le 1/2\). 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.