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• # Artykuł - szczegóły

## Acta Arithmetica

2016 | 174 | 3 | 255-276

## Average r-rank Artin conjecture

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.

255-276

wydano
2016

### Twórcy

autor
• Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, I-00146 Roma, Italy
autor
• Department of Mathematics, Koc University, Rumelifeneri Yolu, 34450 Sarıyer-İstanbul, Turkey