Asymptotic expansions of multiple zeta functions and power mean values of Hurwitz zeta functions

Let zeta(s, alpha) be the Hurwitz zeta function with parameter alpha. Power mean values of the form Sigma(a=1)(q) zeta(s,a/q)(h) or Sigma(a=1)(q) \zeta(s,a/q)\(2h) are studied, where q and h are positive integers. These mean values can be written as a linear combinations of Sigma(a=1)(q) zeta(r)(s(1),.. s(r); a/q), where zeta(r)(s(1),... s(r); alpha) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of Sigma(a=1)(q) zeta(r)(s(1),.. s(r); a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for Sigma(a=1)(q) zeta(s,a/q)(h) and Sigma(a=1)(q) \zeta(s,a/q)\(2h) is obtained. In particular, the asymptotic expansion of Sigma(a=1)(q) zeta(1/2,a/q)(3) with respect to q is written down.