Unified treatment of Artin-type problems II

Abstract

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 $(W_i\mathrm{mod}\mathfrak{p})$. 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.