Zeta Functions for the Adjoint Action of GL(n) and Density of Residues of Dedekind Zeta Functions

Abstract. We define zeta functions for the adjoint action of GLn on its Lie algebra and study their analytic properties. For n ≤ 3 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector space of binary quadratic forms for n = 2. Our construction naturally yields a regularisation, which is necessary for the improvement of the properties of these zeta function, in particular for the analytic continuation if n ≥ 3. We further obtain upper and lower bounds on the mean value X