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

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] $z\notin \mathbb{R}$ and $0<a\le 1$. In addition, we give a new proof of the functional equation of $\Phi (s,a,z)$.

arXiv comment9 pages

References in corpus (1)

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