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.

#### 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 $9\cdot 10^4$.
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Pappalardi F (1997). The r-rank Artin Conjecture. MATHEMATICS OF COMPUTATION, 66(218), 853-868 [10.1090/S0025-5718-97-00805-3].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/143763
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