Zeta functions in several variables associated with prehomogeneous vector spaces, III.Eisenstein series for indefinite quadratic forms

Abstract

geneous vector spaces developed in [ZF-I] and [ZF-II]. In the present paper, we construct certain prehomogeneous vector spaces (p.v.'s of descending chains) related to quadratic forms. Applying the results in [ZF-I] and [ZF-II] to such p.v.'s, we are able to associate with any rational non-degenerate quadratic form with an arbitrary signature a system of Dirichlet series which is a natural generalization of the Eisenstein series and prove that they satisfy certain functional equations. It should be mentioned that Selberg has already pointed out the possibility of such a generalization in [8].