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Let Γ ⊂ Q * be a finitely generated subgroup and let p be a prime such that the reduction group Γ p is a well defined subgroup of the multiplicative group F * p . We prove an asymptotic formula for the average of the number of primes p ≤ x for which the index [F * p : Γ p ] = m. The average is performed over all finitely generated subgroups Γ = a 1 , . . . , a r ⊂ Q * , with a i ∈ Z and a i ≤ T i , with a range of uniformity T i > exp(4(log x log log x) 1 2 ) for every i = 1, . . . , r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar range of uniformity. The case of rank 1 and m = 1 corresponds to the classical Artin's conjecture for primitive roots and has already been considered by Stephens in 1969.

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.

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