ON ZEROS OF BILATERAL HURWITZ AND PERIODIC ZETA AND ZETA STAR FUNCTIONS

Abstract

We show that (1) the periodic zeta function Li ⁡s(e2πia) with 0<a<12 or 12<a<1 does not vanish on the real line; (2) all real zeros of Y(s,a):=ζ(s,a)−ζ(s,1−a), O(s,a):=−iLi ⁡s(e2πia)+iLi ⁡s(e2πi(1−a)) and X(s,a):=Y(s,a)+O(s,a) with 0<a<12 are simple and are located only at the negative odd integers; (3) all real zeros of Z(s,a):=ζ(s,a)+ζ(s,1−a) are simple and are located only at the nonpositive even integers if and only if 14≤a≤12; (4) all real zeros of P(s,a):=Li ⁡s(e2πia)+Li ⁡s(e2πi(1−a)) are simple and are located only at the negative even integers if and only if 14≤a≤12. Moreover, the asymptotic behavior of real zeros of Z(s,a) and P(s,a) are studied when 0<a<14. In addition, the complex zeros of these zeta functions are also discussed when 0<a<12 is rational or transcendental.