## Average $r$-rank Artin conjecture

### Volume 174 / 2016

#### Abstract

Let $\varGamma\subset\mathbb Q^*$ be a finitely generated subgroup and let $p$ be a prime such that the reduction group $\varGamma_p$ is a well defined subgroup of the multiplicative group $\mathbb F_p^*$. We prove an asymptotic formula for the average of the number of primes $p\le x$ for which $[\mathbb F_p^*:\varGamma_p]=m$. The average is taken over all finitely generated subgroups $\varGamma=\langle a_1,\dots,a_r \rangle\subset\mathbb Q^*$, with $a_i\in\mathbb Z$ and $a_i\le T_i$, with a range of uniformity $T_i \gt \exp(4(\log x \log\log x)^{{1}/{2}})$ 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 Artin’s classical conjecture for primitive roots and was already considered by Stephens in 1969.