Shintani zeta function

For the Shintani zeta function of a vector space, see Prehomogeneous vector space.

In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by Takuro Shintani (1976). They include Hurwitz zeta functions, Barnes zeta functions, and Witten zeta functions as special cases.

Definition

The Shintani zeta function of (s₁, ..., s_k) is given by

\sum _{n_{1},\dots ,n_{m}\geq 0}{\frac {1}{L_{1}^{s_{1}}\cdots L_{k}^{s_{k}}}},

where each L_j is an inhomogeneous linear function of (n₁, ... ,n_m). The special case when k = 1 is the Barnes zeta function.

Hida, Haruzo (1993), Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts, 26, Cambridge University Press, ISBN 978-0-521-43411-9, MR 1216135, Zbl 0942.11024
Shintani, Takuro (1976), "On evaluation of zeta functions of totally real algebraic number fields at non-positive integers", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 23 (2): 393–417, ISSN 0040-8980, MR 0427231, Zbl 0349.12007

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