Zeta Functions for the Adjoint Action of GL(n) and density of residues of Dedekind zeta functions

TL;DR

This paper introduces new zeta functions related to the adjoint action of GL(n), analyzes their properties for small n, and derives bounds on residues connected to the distribution of certain number fields.

Contribution

It defines and studies zeta functions for the adjoint action of GL(n), providing analytic properties, regularisation methods, and bounds on residues for number field density estimates.

Findings

Complete analysis for n<4

Recovery of Shintani zeta function for n=2

Bounds on residues related to cubic number fields

Abstract

We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 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>2. We further obtain upper and lower bounds on the mean value X^{-5/2}\sum_{E} \res_{s=1}\zeta_E(s) as X->\infty, 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 GL(3). These asymptotic bounds are a first step towards a generalisation of density results obtained by Datskovsky in case of quadratic field extensions.