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