# Average r-rank Artin conjecture

Acta Arithmetica (2016)

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

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## Abstract

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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 $\Gamma =⟨a₁,...,{a}_{r}⟩\subset ℚ*$, with ${a}_{i}\in ℤ$ and ${a}_{i}\le {T}_{i}$, with a range of uniformity ${T}_{i}>exp\left(4{\left(logxloglogx\right)}^{1/2}\right)$ 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.

## How to cite

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Lorenzo 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 -

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