Shintani zeta function

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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[edit]

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

\sum_{n_1,\dots,n_m\ge 0}\frac{1}{L_1^{s_1} \cdots L_k^{s_k}},

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

References[edit]