Average r-rank artin conjecture

Abstract

Let \Gamma\subset\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^*:\Gamma_p]=m. The average is performed over all finitely generated subgroups \Gamma=\langle a_1,\dots,a_r \rangle\subset\Q^*, with a_i∈Z and a_i≤T_i with a range of uniformity: T_i>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.