Unified treatment of Artin-type problems II

TL;DR

This paper investigates the distribution of primes in number fields related to Artin's conjecture, providing density formulas for preimages of certain index maps under GRH, and expressing these densities via Artin-type constants.

Contribution

It offers a detailed description of the density of primes with prescribed properties in number fields, extending Artin-type problems and expressing densities through new formulas and constants.

Findings

Density of primes can be expressed as a limit and a multiple of Artin-type constants.

Preimages of valuation-defined sets have explicitly computable densities.

Conditional on GRH, the distribution of primes in these contexts is well-understood.

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 $(W_i \bmod \mathfrak p)$. 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.