# Real zeros of Hurwitz-Lerch zeta functions in the interval $(-1,0)$

Research paper by **Takashi Nakamura**

Indexed on: **01 Dec '15**Published on: **01 Dec '15**Published in: **Mathematics - Number Theory**

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#### Abstract

For $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$, 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 $\Phi
(\sigma,a,z) \ne 0$ when $\sigma \in (-1,0)$ if and only if [I] $z=1$ and
$(3-\sqrt{3}) /6 \le a \le 1/2$ or $(3+\sqrt{3}) /6 \le a \le 1$, [II] $z \in
[-1,1)$ and $(1-z)(1-a) \le 1$, [III] $z \not \in {\mathbb{R}}$ and $0<a \le
1$. In addition, we give a new proof of the functional equation of $\Phi
(s,a,z)$.