Functional equation, upper bounds and analogue of Lindel\"of hypothesis for the Barnes double zeta function

TL;DR

This paper derives functional equations for the Barnes double zeta function and uses them to establish upper bounds on its growth, contributing to the understanding of its analytic properties.

Contribution

It introduces functional equations for the Barnes double zeta function and applies convexity principles to obtain growth bounds, extending classical zeta function analysis.

Findings

Functional equations for Barnes double zeta function derived.

Upper bounds for the zeta function established as t approaches infinity.

Application of Phragmén-Lindelöf principle to this context.

Abstract

The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time, and there is great importance in studying these zeta functions. For example, fundamental properties such as the upper bounds, the distribution of zeros, and the zero-free regions in the Riemann zeta function derive from functional equations. In this paper, we consider the functional equations for the Barnes double zeta-function $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $. Additionally, by applying this functional equation and the Phragm\'en-Lindel\"of convexity principle, we obtain some upper bounds for $ \zeta_2(\sigma + it, \alpha ; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.