Tokyo Journal of Mathematics

Mean Values of the Barnes Double Zeta-function


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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 $\text{Im}(s) \rightarrow + \infty$.

Article information

Tokyo J. Math., Volume 41, Number 2 (2018), 557-572.

First available in Project Euclid: 18 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11B06


MIYAGAWA, Takashi. Mean Values of the Barnes Double Zeta-function. Tokyo J. Math. 41 (2018), no. 2, 557--572.

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