Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups , with and , with a range of uniformity 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 Artin’s classical conjecture for primitive roots and was already considered by Stephens in 1969.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8258-4-2016,
author = {Lorenzo Menici and Cihan Pehlivan},
title = {Average r-rank Artin conjecture},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {255-276},
zbl = {06622303},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8258-4-2016}
}
Lorenzo Menici; Cihan Pehlivan. Average r-rank Artin conjecture. Acta Arithmetica, Tome 172 (2016) pp. 255-276. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8258-4-2016/