  Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup
of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm
ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the
multiplicative group of the residue field at $\mathfrak p$. 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.
