Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions: III

Masanori Katsurada, Kohji Matsumoto

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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.

Original languageEnglish
Pages (from-to)239-266
Number of pages28
JournalCompositio Mathematica
Volume131
Issue number3
DOIs
Publication statusPublished - 2002

Keywords

  • Asymptotic expansion
  • Hurwitz zeta function
  • Mean square
  • Riemann zeta function

ASJC Scopus subject areas

  • Algebra and Number Theory

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