Remarks on Shintani's zeta function
Vol. 12 (2005), No. 2, Page 289--317.
Wakayama, Masato
Remarks on Shintani's zeta function
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Abstract:
We introduce a zeta function attached to a representation of a group. We show that the multi-dimensional zeta function due to Shintani [Sh 1], which is a generalization of the multiple Hurwitz zeta function, can be obtained in this framework. We also construct a gamma function from the zeta function attached to a representation via zeta regularization. We study then a $q$-analogue of the Shintani zeta function and the corresponding gamma function. A sine function defined via the reflection formula of such $q$-Shintani gamma function is shown to be a natural generalization of the multiple elliptic function in [Ni]. Moreover, a certain non-commutative group-analogue of the Shintani zeta function is investigated.
Keywords: representation, zeta regularization, multiple Hurwitz zeta function, multiple sine function, multiple elliptic gamma function, multiple Bernoulli polynomial
Mathematics Subject Classification (2000): 11M36
Mathematical Reviews Number: MR2150739
Received: 2004-05-20
