Average $r$-rank Artin conjecture

Abstract

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.