On a density property of the residual order of a (mod pq)

摘要

We consider a distribution property of the residual order (the multiplicative order) of the residue class a(mod pq). It is known that the residual order fluctuates irregularly and increases quite rapidly. We are interested in how the residual orders a (mod pq) distribute modulo 4 when we fix a and let p and q vary. In this paper we consider the set S(ⅹ) = {(p,q);p,q are distinct primes, pq ≤ x}, and calculate the natural density of the set {(p, q) ∈ S(ⅹ); the residual order of a(mod pq) = l(mod 4)}. We show that, under a simple assumption on a, these densities are {5/9, 1/18, 1/3, 1/18} for l = {0,1,2,3}, respectively. For l = 1,3 we need Generalized Riemann Hypothesis.