Asymptotic Expansions of Multiple Zeta Functions and Power Mean Values of Hurwitz Zeta Functions

Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where _r(s₁,...,s_r;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.