Optimal groups for the $r$-rank Artin Conjecture

Abstract

For any finitely generated subgroup $\Gamma$ of $\mathbb{Q}^*$, Pappalardi and the first--named author [1] found a formula to compute the density of the primes $\ell$ for which the reduction modulo $\ell$ of $\Gamma$ contains a primitive root modulo $\ell$. They conjectured a characterization of optimal groups, free or torsion, i.e. subgroups with maximal density. In this paper we prove their conjecture and give a similar characterization for optimal positive groups.