Computing higher rank primitive root densities

TL;DR

This paper generalizes the computation of densities for higher rank primitive roots using an algebraic approach, replacing complex analytic methods with a more conceptual and systematic framework.

Contribution

It extends the character sum method for primitive root densities to radical extensions of arbitrary rank, simplifying calculations and broadening applicability.

Findings

Provides a new algebraic set-up for density calculations

Eliminates lengthy analytic number theory arguments

Enables systematic extension of density formulas

Abstract

We extend the character sum method for the computation of densities in Artin primitive root problems developed by H. W. Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the key parameters of the situation at hand, and obviates the lengthy analytic multiplicative number theory arguments that used to go into the computation of actual densities. It yields a conceptual interpretation of the formulas obtained, and enables us to extend their range of application in a systematic way.