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

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 $t\to \infty$.

19pages