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

@article{Nakamura2015RealZO,
  title={Real zeros of Hurwitz-Lerch zeta functions in the interval \$(-1,0)\$},
  author={T. Nakamura},
  journal={arXiv: Number Theory},
  year={2015}
}
  • T. Nakamura
  • Published 2015
  • Mathematics
  • arXiv: Number Theory
For $0 1$. 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 $\Phi (s,a,z)$. 

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