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December 2018 Mean Values of the Barnes Double Zeta-function
Takashi MIYAGAWA
Tokyo J. Math. 41(2): 557-572 (December 2018). DOI: 10.3836/tjm/1502179261

Abstract

In the study of order estimation of the Riemann zeta-function $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, solving Lindelöf hypothesis is an important theme. As one of the relationships, asymptotic behavior of mean values has been studied. Furthermore, the theory of the mean values is also noted in the double zeta-functions, and the mean values of the Euler-Zagier type of double zeta-function and Mordell-Tornheim type of double zeta-function were studied. In this paper, we prove asymptotic formulas for mean square values of the Barnes double zeta-function $\zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (\alpha+vm+wn)^{-s}$ with respect to $\text{Im}(s)$ as $\text{Im}(s) \rightarrow + \infty$.

Citation

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Takashi MIYAGAWA. "Mean Values of the Barnes Double Zeta-function." Tokyo J. Math. 41 (2) 557 - 572, December 2018. https://doi.org/10.3836/tjm/1502179261

Information

Published: December 2018
First available in Project Euclid: 18 December 2017

zbMATH: 07053492
MathSciNet: MR3908810
Digital Object Identifier: 10.3836/tjm/1502179261

Subjects:
Primary: 11M32
Secondary: 11B06

Rights: Copyright © 2018 Publication Committee for the Tokyo Journal of Mathematics

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Vol.41 • No. 2 • December 2018
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