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

  • Masanori Katsurada (a1) and Kohji Matsumoto (a2)

Abstract

The main object of this paper is the mean square Ih(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for Ih(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for Ih(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 Ih(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.

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Keywords

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

  • Masanori Katsurada (a1) and Kohji Matsumoto (a2)

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