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 . They include Hurwitz zeta functions and Barnes zeta functions. Definition. Let formula_1 be a polynomial in the variables formula_2 with real coefficients such that formula_1 is a product of linear polynomials with positive coefficients, that is, formula_4, where formula_5where formula_6, formula_7 and formula_8. The "Shintani zeta function" in the variable formula_9 is given by (the meromorphic continuation of)formula_10 The multi-variable version. The definition of Shintani zeta function has a straightforward generalization to a  zeta function in several variables formula_11 given byformula_12The special case when "k" = 1 is the Barnes zeta function. Relation to Witten zeta functions. 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 formula_13 defines the Witten zeta function of formula_14 but the linear form formula_15 has formula_16-coefficient equal to zero. Shintani zeta