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

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 ${\mathrm{ind}}_{\mathfrak{p}}(G)$ the index of the subgroup $(G\mathrm{mod}\mathfrak{p})$ 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.

References in corpus (4)

• On the distribution of the order and index of g(mod p) over residue classes II
• On the average number of elements in a finite field with order or index in a prescribed residue class
• On the distribution of the order and index of g(mod p) over residue classes
• Primes in arithmetic progressions and nonprimitive roots