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2023-01-16T04:49:02Z
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ASYMPTOTIC EXPANSIONS OF MULTIPLE ZETA FUNCTIONS AND POWER MEAN VALUES OF HURWITZ ZETA FUNCTIONS
EGAMI, SHIGEKI
MATSUMOTO, KOHJI
open access
Cambridge University Press
Let ζ(s,α) be the Hurwitz zeta function with parameter α. Power mean values of the form ∑^q_a=1ζ(s,α/q)^h or ∑^q_a=1|ζ(s,α/q)|^2h are studied, where q and h are positive integers. These mean values can be written as linear combinations of, ∑^q_a=1 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 ∑^q_a=1ζ_r(s_1,...,s_r;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for ∑^q_a=1ζ(s,α/q)^h and ∑^q_a=1|ζ(s,α/q)|^2h is obtained. In particular, the asymptotic expansion of ∑^q_a=1ζ(1/2,a/q)^3 with respect to q is written down.
Cambridge University Press
2002-08
eng
journal article
VoR
http://hdl.handle.net/2237/10284
https://nagoya.repo.nii.ac.jp/records/8536
https://doi.org/10.1112/S0024610702003253
0024-6107
Journal of the London Mathematical Society
66
1
41
60
https://nagoya.repo.nii.ac.jp/record/8536/files/KOHJI_MATSUMOTO_Asymptotic_Expansions_of_Multiple_Zeta_2002.pdf
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2018-02-19