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Power series and asymptotic series associated with the Lerch zeta-function. (English) Zbl 0937.11035
Let \(s\) be a complex variable, \(\alpha\) and \(\lambda\) real parameters with \(\alpha>0\). We define the Lerch zeta-function \[ \varphi(\lambda, \alpha,s)= \sum_{n=0}^\infty e^{2\pi in\lambda} (n+\alpha)^{-s} \quad (\operatorname {Re}s>1). \] For \(\lambda\in \mathbb{R}\setminus \mathbb{Z}\), this function is continued to an entire function over the \(s\)-plane. If \(\lambda\in \mathbb{Z}\), then this function reduces to the Hurwitz zeta-function \(\zeta(s,\alpha)\). In this paper, based on the Mellin-Barnes type of integral formulae the author studies the power series and asymptotic series for the Lerch zeta-function \(\varphi(\lambda, \alpha,s)\) in the second parameter, and obtains several asymptotic formulae.

11M35 Hurwitz and Lerch zeta functions
Full Text: DOI
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