On the estimation of the order of Euler-Zagier multiple zeta-functions

Abstract

We prove upper bound estimates for Euler-Zagier multiple zeta-functions. First, by shifting the paths of the relevant Mellin-Barnes type integrals to the right, we prove an estimate for general $r$-fold zeta-functions. Then, in the cases $r=2$ and $r=3$, we give further improvements by shifting the path suitably to the left.