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 \(\mathop{\mathrm{GL}}\nolimits _{n}\) on its Lie algebra and study their analytic properties. For n ≤ 3 we are able to fully analyse these functions. If n = 2, we recover the Shintani zeta function for the prehomogeneous vector space of binary quadratic forms. Our construction naturally yields a regularisation, which is necessary to improve the analytic 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 \(X^{-\frac{5} {2} }\sum _{E}\mathop{ \mathrm{res}}_{s=1}\zeta _{E}(s)\) as X → ∞, where E runs over totally real cubic number fields whose second successive minimum of the trace form on its ring of integers is bounded by X. To prove the upper bound we use our new zeta function for \(\mathop{\mathrm{GL}}\nolimits _{3}\). These asymptotic bounds are a first step towards a generalisation of density results obtained by Datskovsky in case of quadratic field extensions.