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

Published online by Cambridge University Press:  12 October 2015

TAKASHI NAKAMURA
TAKASHI NAKAMURA*
Affiliation:
Department of Liberal Arts, Faculty of Science and Technology, Tokyo University of Science 2641 Yamazaki, Noda-shi, Chiba-ken, 278-8510, Japan. e-mail: nakamuratakashi@rs.tus.ac.jp
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Abstract

Let 0 < a ⩽ 1, s, z ∈ ${\mathbb{C}}$ and 0 < |z| ⩽ 1. Then the Hurwitz–Lerch zeta function is defined by Φ(s, a, z) ≔ ∑∞n = 0zn(n + a)− s when σ ≔ ℜ(s) > 1. In this paper, we show that the Hurwitz zeta function ζ(σ, a) ≔ Φ(σ, a, 1) does not vanish for all 0 < σ < 1 if and only if a ⩾ 1/2. Moreover, we prove that Φ(σ, a, z) ≠ 0 for all 0 < σ < 1 and 0 < a ⩽ 1 when z ≠ 1. Real zeros of Hurwitz–Lerch type of Euler–Zagier double zeta functions are studied as well.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 39 - 50
Copyright
Copyright © Cambridge Philosophical Society 2015 

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