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arXiv · 1704.01850

Approximate functional equations for the Hurwitz and Lerch zeta-functions

Takashi Miyagawa
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Abstract

As one of the asymptotic formulas for the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In 2003, R. Garunkštis, A. Laurinčikas, and J. Steuding (in [1]) proved the Riemann-Siegel type of the approximate functional equation for the Lerch zeta-function $ ζ_L (s, α, λ) = \sum_{n=0}^\infty e^{2πi n λ}(n + α)^{-s} $. In this paper, we prove another type of approximate functional equations for the Hurwitz and Lerch zeta-functions. R. Garunkštis, A. Laurinčikas, and J. Steuding (in \cite{GLS2}) obtained the results on the mean square values of $ ζ_L (σ+ it, α, λ) $ with respect to $ t $. We obtain the main term of the mean square values of $ ζ_L (1/2 + it, α, λ) $ using a simpler method than their method in [2].

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Takashi Miyagawa. 2017-04-06. Approximate functional equations for the Hurwitz and Lerch zeta-functions. https://arxiv.org/abs/1704.01850

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