Given Γ⊂Q∗ a multiplicative subgroup and m∈N+ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind p Γ = m, where ind p Γ = (p − 1)/|Γ p | and Γ p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations.

Pappalardi F, & Susa A (2013). An analogue of Artin’s conjecture for multiplicative subgroups of the rationals. ARCHIV DER MATHEMATIK, 101(4), 319-330 [10.1007/s00013-013-0563-7].

An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

PAPPALARDI, FRANCESCO;
2013

Abstract

Given Γ⊂Q∗ a multiplicative subgroup and m∈N+ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind p Γ = m, where ind p Γ = (p − 1)/|Γ p | and Γ p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations.
Pappalardi F, & Susa A (2013). An analogue of Artin’s conjecture for multiplicative subgroups of the rationals. ARCHIV DER MATHEMATIK, 101(4), 319-330 [10.1007/s00013-013-0563-7].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/144585
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