Masanori Katsurada

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 .