Average r-rank Artin conjecture

• Autorzy: Lorenzo Menici , Cihan Pehlivan
• Języki publikacji: EN

Abstrakty

EN

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 𝔽ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [𝔽ₚ*: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.

Kategorie tematyczne

• 11L40: Estimates on character sums
• 11N69: Distribution of integers in special residue classes
• 11A07: Congruences; primitive roots; residue systems
• 11R45: Density theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 174
• Numer: 3
• Strony: 255-276

Daty

wydano

2016

Twórcy

autor

Lorenzo Menici

• Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, I-00146 Roma, Italy

autor

Cihan Pehlivan

• Department of Mathematics, Koc University, Rumelifeneri Yolu, 34450 Sarıyer-İstanbul, Turkey

Identyfikatory

DOI

10.4064/aa8258-4-2016