A Generalization of the Siegel–Walfisz Theorem and its Application

Abstract

We investigate the distribution of integers with a fixed number of prime factors in arithmetic progressions, and obtain a generalization of the Siegel–Walfisz theorem under the extended Riemann hypothesis. As an application, we consider a problem of P. Erdős, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo q by products of two integers with a fixed number of prime factors. We show some conditional results.