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

Let ζ (s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form ∑_a=1^qζ(s,a/q)^h or ∑_a=1^qζ(s,a/q)^2h are studied, where q and h are positive integers. These mean values can be written as linear combinations of ∑_a=1^qζ_r(S₁,., S_r;a/q), where ζ_r(S₁., S_r; α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of ∑_a=1^qζ_r(S₁,., S_r,;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for ∑_a=1^qζ(s, a/q)^h and ∑_a=1^qζ(s, a/q)^2h is obtained. In particular, the asymptotic expansion of ∑_a=1^qζ(1/2, a/q)³ with respect to q is written down.