Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions: III

Abstract

The main object of this paper is the mean square I_h(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for I_h(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for I_h(1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for I_h(1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.