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Average $r$-rank Artin conjecture

Volume 174 / 2016

Lorenzo Menici, Cihan Pehlivan Acta Arithmetica 174 (2016), 255-276 MSC: Primary 11R45; Secondary 11N69, 11A07, 11L40. DOI: 10.4064/aa8258-4-2016 Published online: 22 June 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.

Authors

  • Lorenzo MeniciDipartimento di Matematica
    Università Roma Tre
    Largo S. L. Murialdo, 1
    I-00146 Roma, Italy
    e-mail
  • Cihan PehlivanDepartment of Mathematics
    Koc University
    Rumelifeneri Yolu
    34450 Sarıyer-İstanbul, Turkey
    e-mail

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