Répartition Des Diviseurs Dans Les Progressions Arithmétiques

Abstract

Let q and N be integers, let a be an integer coprime to q, and let zN be defined implicitly by q = ( log N ) log 2 2 − z N √ ( log 2 N ) . We show that for large N, an integer n has at least one divisor d with q ⩽ d ⩽ N and d ≡ a(mod q) with probability approximately Φ(zN), where Φ denotes the distribution function of the Gaussian Law. This solves a conjecture of Hall. 1991 Mathematics Subject Classification 11N25, 11N37.