Average $r$-rank Artin conjecture

Abstract

Let ⊂ Qbe a finitely generated subgroup and let p be a prime such that the reduction group p is a well defined subgroup of the multiplicative group F �. We prove an asymptotic formula for the average of the number of primes p ≤ x for which the index (F � : p) = m. The average is performed over all finitely generated subgroups = ha1,...,ari ⊂ Q � , with ai ∈ Z and ai ≤ Ti with a range of uniformity: Ti > exp(4(log xlog 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 conjecture for primitive roots and has already been considered by Stephens in 1969.