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

TL;DR

This paper characterizes the conditions under which the Hurwitz-Lerch zeta function has zeros in the interval (-1,0) for complex parameters, and provides a new proof of its functional equation.

Contribution

It precisely describes the zero-free regions of the Hurwitz-Lerch zeta function in the interval (-1,0) and offers a new proof of its functional equation.

Findings

Zeros are absent in specified parameter regions.

Explicit conditions for zeros when z=1.

New proof of the functional equation.

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 $\Phi (s,a,z)$.