Mean values of the Barnes double zeta-function

TL;DR

This paper derives asymptotic formulas for the mean square values of the Barnes double zeta-function as the imaginary part of the complex variable grows large, extending mean value theory to this special function.

Contribution

It provides the first asymptotic formulas for the mean square values of the Barnes double zeta-function, expanding the understanding of its mean value behavior.

Findings

Asymptotic formulas for mean square values as Im(s) → ∞

Extension of mean value theory to Barnes double zeta-function

Results contribute to the broader study of zeta-function mean values

Abstract

In the study of order estimation of the Riemann zeta-function $ \zeta(s) = \sum_{n=1}^\infty n^{-s} $, solving Lindel\"{o}f 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 $ \mathrm{Im}(s) $ as $ \mathrm{Im}(s) \rightarrow + \infty $.