RÉPARTITION DES DIVISEURS DANS LES PROGRESSIONS ARITHMÉTIQUES

Abstract

Let q and N be integers, let a be an integer coprime to q, and let z_N be defined implicitly by q = (log N)^log22^{−Z_N √(log₂N)}. We show that for large N, an integer n has at least one divisor d with q [les ] d [les ] N and d ≡ a(mod q) with probability approximately Φ(z_N), where Φ denotes the distribution function of the Gaussian Law. This solves a conjecture of Hall.