Average r-rank Artin Conjecture

2015/04/07 by Cihan Pehlivan, Pehlivan, Cihan, Lorenzo Menici +1
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1504.01554

openalex publication_date 2015/04/07 · arxiv created 2015/08/12 · arxiv updated 2015/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ⊂ℚ^* be a finitely generated subgroup and let p be a prime such that the reduction group Γ_p is a well defined subgroup of the multiplicative group \mathbbF_p^*. We prove an asymptotic formula for the average of the number of primes p≤ x for which the index [\mathbbF_p^*:Γ_p]=m. The average is performed 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 the classical Artin conjecture for primitive roots and has already been considered by Stephens in 1969.

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