Real zeros of Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta functions

Abstract

Let $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$. Then the Hurwitz-Lerch zeta function is defined by $\Phi (s,a,z) := \sum_{n=0}^\infty z^n(n+a)^{-s}$ when $\sigma :=\Re (s) >1$. In this paper, we show that the Hurwitz zeta function $\zeta (\sigma,a) := \Phi (\sigma,a,1)$ does not vanish for all $0 <\sigma <1$ if and only if $a \ge 1/2$. Moreover, we prove that $\Phi (\sigma,a,z) \ne 0$ for all $0 <\sigma <1$ and $0 < a \le 1$ when $z \ne 1$. Real zeros of Hurwitz-Lerch type of Euler-Zagier double zeta functions are studied as well.