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

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.