The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I

We consider general multiple zeta-functions of multi-variables, including both Barnes multiple zeta-functions and Euler-Zagier sums as special cases. We prove the meromorphic continuation to the whole space, asymptotic expansions, and upper bound estimates. These results are expected to have applications to some arithmetical L-functions (such as of Hecke and of Shintani). The method is based on the classical Mellin-Barnes integral formula.