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

TL;DR

This paper establishes an approximate functional equation for the Barnes double zeta-function and uses it to derive upper bounds for its growth along the critical line as the imaginary part tends to infinity.

Contribution

It introduces a new approximate functional equation for the Barnes double zeta-function and applies it to obtain bounds on its size for large imaginary parts.

Findings

Derived an approximate functional equation for Barnes double zeta-function.

Established upper bounds for the zeta-function on the critical line.

Applied the van der Corput method to obtain growth estimates.

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 $.