Congruence properties of the Omega-function on sumsets

In this article we investigate the behaviour of the omega function, which c ounts the number of prime factors of an integer with multiplicity, as one r uns over those integers of the form a + b where a is from a set A and b is from a set B. We prove, for example, that if A and B are sufficiently dense subsets of the first N positive integers and k is a positive integer then the number of pairs (a, b) for which the omega function of a + b lies in a given residue class module k is roughly the total number of pairs divided b y k.