On the distribution of the order over residue classes

Pieter Moree
2006 Electronic Research Announcements of the American Mathematical Society  
For a fixed rational number g ∈ {−1, 0, 1} and integers a and d we consider the set N g (a, d) of primes p such that the order of g modulo p is congruent to a (mod d). Under the Generalized Riemann Hypothesis (GRH), it can be shown that the set N g (a, d) has a natural density δ g (a, d). Arithmetical properties of δ g (a, d) are described, and δ g (a, d) is compared with δ(a, d): the average density of elements in a field of prime characteristic having order congruent to a (mod d). It
more » ... mod d). It transpires that δ g (a, d) has a strong tendency to be equal to δ(a, d), or at least to be close to it.
doi:10.1090/s1079-6762-06-00168-5 fatcat:5etr75j25rcodbt2egzhmjj2xy