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