## Remarks on Shintani's zeta function

J. Math. Sci. Univ. Tokyo
Vol. 12 (2005), No. 2, Page 289--317.

Wakayama, Masato
Remarks on Shintani's zeta function
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.