A two-variable Artin conjecture

[摘要] Let a, b is an element of Q* be rational numbers that are multiplicatively independent. We study the natural density delta (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 delta (a, b) exists and equals a positive rational multiple of the universal constant S = Pi (p prime) (1 - p/(p(3) - 1)). An explicit value of delta (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 delta (a, b). (C) 2000 Academic Press.

[发布日期] 2000-12-01

[关键词] Artin's conjecture;primitive roots;recurrence