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arXiv · 1810.08295

Approximate functional equation and upper bounds for the Barnes double zeta-function

Takashi Miyagawa
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Abstract

As one of the asymptotic formulas of the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In this paper, we prove an approximate functional equation of the Barnes double zeta-function $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $. Also, applying this approximate functional equation and the van der Corput method, we obtain upper bounds for $ \zeta_2(1/2 + it, \alpha ; v, w) $ and $ \zeta_2(3/2 + it, \alpha ; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.

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Takashi Miyagawa. 2018-10-18. Approximate functional equation and upper bounds for the Barnes double zeta-function. https://arxiv.org/abs/1810.08295

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