On the estimation of the order of euler-zagier multiple zeta-functions

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.