Average Multiplicative Order of Finitely Generated Subgroup of Rational Numbers Over Primes

TL;DR

This paper derives an asymptotic formula for the average multiplicative order of finitely generated subgroups of rational numbers modulo primes, under the Generalized Riemann Hypothesis, extending previous rank 1 results.

Contribution

It provides a general asymptotic formula for the average order over primes for finitely generated subgroups, including explicit densities for positive elements, advancing understanding beyond rank 1 cases.

Findings

Asymptotic formula for average order under GRH

Explicit density expression for positive subgroups

Numerical computations supporting theoretical results

Abstract

Given a finitely generated multiplicative subgroup of rational numbers $\Gamma$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group modulo $p$. The problem was considered in the case of rank $1$ by Pomerance and Kurlberg. In the case when $\Gamma$ contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.