The distribution of the multiplicative index of algebraic numbers over residue classes

Let K be a number field and G a finitely generated torsion-free subgroup of \(K^\times \) $K^{\times }$ . Given a prime \(\mathfrak {p}\) $\mathfrak{p}$ of K we denote by \({{\,\textrm{ind}\,}}_\mathfrak {p}(G)\) ${\text{ind}}_{\mathfrak{p}}(G)$ the index of the subgroup \((G\bmod \mathfrak {p})\) $(Gmod\mathfrak{p})$ of the multiplicative group of the residue field at \(\mathfrak {p}\) $\mathfrak{p}$ . Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.