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

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 $.