Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta-function

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 are great importance when studying these zeta-functions. For example, fundamental properties of the upper bounds, the distribution of zeros, the zero-free regions in the Riemann zeta function start from functional equations. In this paper, we prove a functional equations 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 functional equation and the Phragmén-Lindel\"of convexity principle, we obtain some upper bounds for \( \zeta_2(\sigma + it, \alpha ; v, w) \ (0\leq \sigma \leq 2) \) with respect to \( t \) as \( t \rightarrow \infty \).