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Let $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$. Then theHurwitz-Lerch zeta function is defined by $\Phi (s,a,z) := \sum_{n=0}^\inftyz^n(n+a)^{-s}$ when $\sigma :=\Re (s) >1$. In this paper, we show that theHurwitz zeta function $\zeta (\sigma,a) := \Phi (\sigma,a,1)$ does not vanishfor 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 $z \ne1$. Real zeros of Hurwitz-Lerch type of Euler-Zagier double zeta functions arestudied as well.

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