Unconditional results for Artin-type problems over number fields

TL;DR

This paper investigates the distribution of primes in number fields where the reduction of a finitely generated subgroup of the field's multiplicative group has a specified index, providing unconditional results and asymptotic formulas without relying on GRH.

Contribution

It establishes unconditional results for Artin-type problems over number fields, extending previous work that depended on GRH, and derives asymptotic formulas for prime distributions.

Findings

Proves that certain prime distribution problems can be addressed without GRH under convergence conditions.

Provides asymptotic formulas for prime-counting functions related to subgroup indices.

Extends Artin's primitive root conjecture results to broader settings in number fields.

Abstract

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group of the residue field at $\mathfrak p$, and we may consider its index. We study the primes of $K$ for which this index lies in a given set of positive integers $S$. In particular, we prove that under certain convergence conditions on series associated to $S$ this problem can be addressed without assuming the Generalized Riemann Hypothesis (GRH), and we provide asymptotic formulas for the corresponding prime-counting functions. Problems of this type are related to Artin's primitive root conjecture, which has been proven under the assumption of GRH (Hooley, 1967).