Average r-rank Artin conjecture

Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group 𝔽ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [𝔽ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups $Γ =⟨a₁,...,a_{r}⟩⊂ ℚ *$, with $a_{i} ∈ ℤ$ and $a_{i} ≤ T_{i}$, with a range of uniformity $T_{i} > exp(4(log x loglog x)^{1/2})$ 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 Artin's classical conjecture for primitive roots and was already considered by Stephens in 1969.