Average $r$-rank Artin conjecture

Lorenzo Menici¹ ; Cihan Pehlivan²

¹Dipartimento di Matematica Università Roma Tre Largo S. L. Murialdo, 1 I-00146 Roma, Italy
²Department of Mathematics Koc University Rumelifeneri Yolu 34450 Sarıyer-İstanbul, Turkey

Acta Arithmetica, Tome 174 (2016) no. 3, pp. 255-276

Résumé

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.

Détail

DOI : 10.4064/aa8258-4-2016

Keywords: vargamma subset mathbb * finitely generated subgroup prime reduction group vargamma defined subgroup multiplicative group mathbb * prove asymptotic formula average number primes which mathbb * vargamma average taken finitely generated subgroups vargamma langle dots rangle subset mathbb * mathbb range uniformity exp log log log every dots prove asymptotic formula mean square error terms asymptotic formula similar range uniformity rank corresponds artin classical conjecture primitive roots already considered stephens