### Abstract

Let ζ (s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form ∑_{a=1}^{q}ζ(s,a/q)^{h} or ∑_{a=1}^{q}|ζ(s,a/q)|^{2h} are studied, where q and h are positive integers. These mean values can be written as linear combinations of ∑_{a=1}^{q}ζ_{r}(S_{1},..., S_{r};a/q), where ζ_{r}(S_{1}..., S_{r}; α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of ∑_{a=1}^{q}ζ_{r}(S_{1},..., S_{r},;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for ∑_{a=1}^{q}ζ(s, a/q)^{h} and ∑_{a=1}^{q}|ζ(s, a/q)|^{2h} is obtained. In particular, the asymptotic expansion of ∑_{a=1}^{q}ζ(1/2, a/q)^{3} with respect to q is written down.

Original language | English |
---|---|

Pages (from-to) | 41-60 |

Number of pages | 20 |

Journal | Journal of the London Mathematical Society |

Volume | 66 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 Aug |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the London Mathematical Society*,

*66*(1), 41-60. https://doi.org/10.1112/S0024610702003253

**Asymptotic expansions of multiple zeta functions and power mean values of Hurwitz zeta functions.** / Egami, Shigeki; Matsumoto, Kohji.

Research output: Contribution to journal › Article

*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 41-60. https://doi.org/10.1112/S0024610702003253

}

TY - JOUR

T1 - Asymptotic expansions of multiple zeta functions and power mean values of Hurwitz zeta functions

AU - Egami, Shigeki

AU - Matsumoto, Kohji

PY - 2002/8

Y1 - 2002/8

N2 - Let ζ (s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form ∑a=1qζ(s,a/q)h or ∑a=1q|ζ(s,a/q)|2h are studied, where q and h are positive integers. These mean values can be written as linear combinations of ∑a=1qζr(S1,..., Sr;a/q), where ζr(S1..., Sr; α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of ∑a=1qζr(S1,..., Sr,;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for ∑a=1qζ(s, a/q)h and ∑a=1q|ζ(s, a/q)|2h is obtained. In particular, the asymptotic expansion of ∑a=1qζ(1/2, a/q)3 with respect to q is written down.

AB - Let ζ (s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form ∑a=1qζ(s,a/q)h or ∑a=1q|ζ(s,a/q)|2h are studied, where q and h are positive integers. These mean values can be written as linear combinations of ∑a=1qζr(S1,..., Sr;a/q), where ζr(S1..., Sr; α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of ∑a=1qζr(S1,..., Sr,;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for ∑a=1qζ(s, a/q)h and ∑a=1q|ζ(s, a/q)|2h is obtained. In particular, the asymptotic expansion of ∑a=1qζ(1/2, a/q)3 with respect to q is written down.

UR - http://www.scopus.com/inward/record.url?scp=0036689533&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036689533&partnerID=8YFLogxK

U2 - 10.1112/S0024610702003253

DO - 10.1112/S0024610702003253

M3 - Article

AN - SCOPUS:0036689533

VL - 66

SP - 41

EP - 60

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 1

ER -