On zeros of bilateral Hurwitz and periodic zeta and zeta star functions

Abstract

In this paper, we show the following; (1) The periodic zeta function ${\mathrm{Li}}_s(e^{2\pi ia})$ with $0<a<1/2$ or $1/2<a<1$ does not vanish on the real line. (2) All real zeros of $Y(s,a):=\zeta (s,a)-\zeta (s,1-a)$, $O(s,a):=-i{\mathrm{Li}}_s(e^{2\pi ia})+i{\mathrm{Li}}_s(e^{2\pi i(1-a)})$ and $X(s,a):=Y(s,a)+O(s,a)$ with $0<a<1/2$ are simple and only at the negative odd integers. (3) All real zeros of $Z(s,a):=\zeta (s,a)+\zeta (s,1-a)$ are simple and only at the non-positive even integers if and only if $1/4\le a\le 1/2$. (4) All real zeros of $P(s,a):={\mathrm{Li}}_s(e^{2\pi ia})+{\mathrm{Li}}_s(e^{2\pi i(1-a)})$ are simple and only at the negative even integers if and only if $1/4\le a\le 1/2$. Moreover, the asymptotic behavior of real zeros of $Z(s,a)$ and $P(s,a)$ are studied when $0<a<1/4$. In addition, the complex zeros of these zeta functions are also discussed when $0<a<1/2$ is rational or transcendental.

16 pages. Sections 3.1 and 4.1 are rewritten. Some equations and sentences are also rewritten

References in corpus (4)

• Real zeros of quadratic Dirichlet L-functions
• Zeros of Dirichlet series with periodic coefficients
• The values of zeta functions composed by the Hurwitz and periodic zeta functions at integers
• Functional equation and zeros on the critical line of the quadrilateral zeta function