The average multiplicative order of a finitely generated subgroup of the rationals modulo primes

Abstract

Given a finitely generated multiplicative subgroup Γ ⊆ Q * , 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 Γ 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.