Average $r$-rank Artin Conjecture

Let $\mathrm{\Gamma }\subset {\mathbb{Q}}^{\ast }$ be a finitely generated subgroup and let $p$ be a prime such that the reduction group ${\mathrm{\Gamma }}_p$ is a well defined subgroup of the multiplicative group ${\mathbb{F}}_p^{\ast }$. We prove an asymptotic formula for the average of the number of primes $p\le x$ for which the index $[{\mathbb{F}}_p^{\ast }:{\mathrm{\Gamma }}_p]=m$. The average is performed over all finitely generated subgroups $\mathrm{\Gamma }=⟨a_1,\dots ,a_r⟩\subset {\mathbb{Q}}^{\ast }$, with $a_i\in \mathbb{Z}$ and $a_i\le T_i$ with a range of uniformity: $T_i>\exp (4(\log x\log \log x{)}^{\frac{1}{2}})$ for every $i=1,\dots ,r$. 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.