### 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 |
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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 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the London Mathematical Society*,

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