A Two-Variable Artin Conjecture

Abstract Let 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 ( a  mod  p ) contains ( b  mod  p ). 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 =∏ p  prime  (1− p /( p 3 −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 Theory 8 , 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 ).