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

Abstract

Let 0 < a ≤ 1/2 and define the quadrilateral zeta function by 2Q(s, a) := ζ(s, a)+ ζ(s, 1 − a)+ Li s (e 2πia )+ Li s (e 2πi(1−a) ), where ζ(s, a) is the Hurwitz zeta function and Li s (e 2πia ) is the periodic zeta function.In the present paper, we show that there exists a unique real number a 0 ∈ (0, 1/2) such that Q(σ, a 0 ) has a unique double real zero at σ = 1/2 when σ ∈ (0, 1), for any a ∈ (a 0 , 1/2], the function Q(σ, a) has no zero in the open interval σ ∈ (0, 1) and for any a ∈ (0, a 0 ), the function Q(σ, a) has at least two real zeros in σ ∈ (0, 1).Moreover, we prove that Q(s, a) has infinitely complex zeros in the region of absolute convergence and the critical strip when a ∈ Q ∩ (0, 1/2) \ {1/6, 1/4, 1/3}. The Riemannvon Mangoldt formula for Q(s, a) is also shown.Theorem A. All real zeros of ζ(s) are simple and at only the negative even integers.