Unconditional results for Artin-type problems over number fields

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^{\times }$. For all but finitely many primes $\mathfrak{p}$ of $K$, the reduction $(G\mathfrak{p})$ generates a well-defined subgroup of the multiplicative group of the residue field at $\mathfrak{p}$, and we may consider its index. We study the primes of $K$ for which this index lies in a given set of positive integers $S$. In particular, we prove that under certain convergence conditions on series associated to $S$ this problem can be addressed without assuming the Generalized Riemann Hypothesis (GRH), and we provide asymptotic formulas for the corresponding prime-counting functions. Problems of this type are related to Artin’s primitive root conjecture, which has been proven under the assumption of GRH (Hooley, 1967).