On the distribution of the order and index of g(mod p) over residue classes

Abstract

For a fixed rational number g, not equal to -1,0 or 1 and integers a and d we consider the set of primes p for which the order of g(mod p) is congruent to a(mod d). For d=4 and d=3 it is shown that, under the Generalized Riemann Hypothesis, these sets have a natural density. Moreover, we explicitly compute this density. For d=4 this generalises earlier work by K. Chinen and L. Murata. The case d=3 was apparently not considered before.