# Average r-rank Artin conjecture

Lorenzo Menici; Cihan Pehlivan

Acta Arithmetica (2016)

- Volume: 174, Issue: 3, page 255-276
- ISSN: 0065-1036

## Access Full Article

top## Abstract

top## How to cite

topLorenzo Menici, and Cihan Pehlivan. "Average r-rank Artin conjecture." Acta Arithmetica 174.3 (2016): 255-276. <http://eudml.org/doc/286473>.

@article{LorenzoMenici2016,

abstract = {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 $Γ =⟨a₁,...,a_\{r\}⟩⊂ ℚ *$, with $a_\{i\} ∈ ℤ$ and $a_\{i\} ≤ T_\{i\}$, with a range of uniformity $T_\{i\} > exp(4(log x loglog x)^\{1/2\})$ 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.},

author = {Lorenzo Menici, Cihan Pehlivan},

journal = {Acta Arithmetica},

keywords = {artin's conjecture; primitive roots; multiple Ramanujan sum},

language = {eng},

number = {3},

pages = {255-276},

title = {Average r-rank Artin conjecture},

url = {http://eudml.org/doc/286473},

volume = {174},

year = {2016},

}

TY - JOUR

AU - Lorenzo Menici

AU - Cihan Pehlivan

TI - Average r-rank Artin conjecture

JO - Acta Arithmetica

PY - 2016

VL - 174

IS - 3

SP - 255

EP - 276

AB - 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 $Γ =⟨a₁,...,a_{r}⟩⊂ ℚ *$, with $a_{i} ∈ ℤ$ and $a_{i} ≤ T_{i}$, with a range of uniformity $T_{i} > exp(4(log x loglog x)^{1/2})$ 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.

LA - eng

KW - artin's conjecture; primitive roots; multiple Ramanujan sum

UR - http://eudml.org/doc/286473

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.