A Two-Variable Artin Conjecture

AbstractLet a, b∈Q* be rational numbers that are multiplicatively independent. We study the natural density δ(a, b) of the set of primes p for which the subgroup of F*p generated by (amodp) contains (bmodp). It is shown that, under assumption of the generalized Riemann hypothesis, the density δ(a, b) exists and equals a positive rational multiple of the universal constant S=∏pprime(1−p/(p3−1)). An explicit value of δ(a, b) is given under mild conditions on a and b. This extends and corrects earlier work of Stephens (1976, J. Number Theory8, 313–332). We also discuss the relevance of the result in the context of second order linear recurrent sequences and some numerical aspects of the determination of δ(a, b)