This work concerns Artin’s Conjecture on primitive roots and related problems for number fields. Let K be a number field and let \(W_1\) to \(W_n\) be finitely generated subgroups of \(K^\times \) of positive rank. We consider the index map, which maps a prime \(\mathfrak p\) of K to the n-tuple of the indices of \((W_i \bmod \mathfrak {p})\) . Conditionally under GRH, any preimage under the index map admits a density, and the aim of this work is describing it. For example, we express the density as a limit in various ways. We study in particular the preimages of sets of n-tuples that are defined by prescribing valuations for their entries. Under some mild assumptions we can express the density as a multiple of a (suitably defined) Artin-type constant.
