On the $r$-rank Artin Conjecture

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 $9\cdot 10^4$.