The distribution of the multiplicative index of algebraic numbers over residue classes

TL;DR

This paper investigates the distribution of the multiplicative index of algebraic numbers modulo primes in number fields, under GRH, focusing on the density of primes with specific index properties and Frobenius elements.

Contribution

It provides a formula for the natural density of primes with given multiplicative index and Frobenius conditions, especially for indices in arithmetic progressions, assuming GRH.

Findings

Derived explicit density formulas under GRH.

Analyzed positivity of densities for arithmetic progression indices.

Provided detailed case studies for specific index sets.

Abstract

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.