### 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.

Original language | English |
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Pages (from-to) | 239-266 |

Number of pages | 28 |

Journal | Compositio Mathematica |

Volume | 131 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 Dec 1 |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Compositio Mathematica*,

*131*(3), 239-266. https://doi.org/10.1023/A:1015585314625