Density of the "quasi $r$-rank Artin problem"

TL;DR

This paper investigates the density of primes related to the reduction of finitely generated subgroups of rationals, providing formulas under GRH, classifying certain cases, and comparing theoretical predictions with computational data.

Contribution

It derives density formulas for primes in the quasi r-rank Artin problem under GRH and classifies rank one torsion groups with vanishing density, offering new insights into prime distribution.

Findings

Formulas for prime densities under GRH

Classification of rank one torsion groups with zero density

Comparison of theoretical and computational densities

Abstract

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value. We completely classify the cases of rank one torsion groups for which the density vanishes and the the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we propose some partial results. Finally, we propose some computations of examples comparing the approximated density computed with primes up to $10^{10}$ and that predicted by the Riemann Hypothesis.