We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which $\mathbb{F}^\ast_p$ can be generated by $r$ given multiplicatively independent numbers. In the case when the $r$ given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown-Zassenhaus (J. Number Theory 3 (1971), 306-309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to $9\cdot 10^4$.
Pappalardi, F. (1997). The r-rank Artin Conjecture. MATHEMATICS OF COMPUTATION, 66(218), 853-868 [10.1090/S0025-5718-97-00805-3].
The r-rank Artin Conjecture.
PAPPALARDI, FRANCESCO
1997-01-01
Abstract
We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which $\mathbb{F}^\ast_p$ can be generated by $r$ given multiplicatively independent numbers. In the case when the $r$ given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown-Zassenhaus (J. Number Theory 3 (1971), 306-309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to $9\cdot 10^4$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
