### 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 |
---|---|

Pages (from-to) | 239-266 |

Number of pages | 28 |

Journal | Compositio Mathematica |

Volume | 131 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

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

**Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions : III.** / Katsurada, Masanori; Matsumoto, Kohji.

Research output: Contribution to journal › Article

*Compositio Mathematica*, vol. 131, no. 3, pp. 239-266. https://doi.org/10.1023/A:1015585314625

}

TY - JOUR

T1 - Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions

T2 - III

AU - Katsurada, Masanori

AU - Matsumoto, Kohji

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Asymptotic expansion

KW - Hurwitz zeta function

KW - Mean square

KW - Riemann zeta function

UR - http://www.scopus.com/inward/record.url?scp=0036277265&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036277265&partnerID=8YFLogxK

U2 - 10.1023/A:1015585314625

DO - 10.1023/A:1015585314625

M3 - Article

AN - SCOPUS:0036277265

VL - 131

SP - 239

EP - 266

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 3

ER -