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

Let $0 < a \le 1/2$ and define the quadrilateral zeta function by $2Q(s,a) := ζ(s,a) + ζ(s,1-a) + {\rm{Li}}_s (e^{2πia}) + {\rm{Li}}_s(e^{2πi(1-a)})$, where $ζ(s,a)$ is the Hurwitz zeta function and ${\rm{Li}}_s (e^{2πia})$ 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 $Q(σ, a_0)$ has a unique double real zero at $σ= 1/2$ when $σ\in (0,1)$, for any $a \in (a_0,1/2]$, the function $Q(σ, a)$ has no zero in the open interval $σ\in (0,1)$ and for any $a \in (0,a_0)$, the function $Q(σ, a)$ has at least two real zeros in $σ\in (0,1)$. 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.

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