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

Abstract

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.