Shintani zeta function
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 and Barnes zeta functions.
Let be a polynomial in the variables with real coefficients such that is a product of linear polynomials with positive coefficients, that is, , where
The multi-variable versionEdit
The definition of Shintani zeta function has a straightforward generalization to a zeta function in several variables given by
Relation to Witten zeta functionsEdit
Just like Shintani zeta functions, Witten zeta functions are defined by polynomials which are products of linear forms with non-negative coefficients. Witten zeta functions are however not special cases of Shintani zeta functions because in Witten zeta functions the linear forms are allowed to have some coefficients equal to zero. For example, the polynomial defines the Witten zeta function of but the linear form has -coefficient equal to zero.
- 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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