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.

arXiv comment12 pages. Some typos are corrected. Reference [7] and Acknowledgments are added

Cited by in corpus (2)

• Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta distributions
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