## Illinois Journal of Mathematics

### Congruence properties of the $\Omega$-function on sumsets

#### Abstract

In this article we investigate the behaviour of the omega function, which counts the number of prime factors of an integer with multiplicity, as one runs 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 modulo $k$ is roughly the total number of pairs divided by $k$.

#### Article information

Source
Illinois J. Math., Volume 43, Issue 1 (1999), 1-18.

Dates
First available in Project Euclid: 19 October 2009

https://projecteuclid.org/euclid.ijm/1255985334

Digital Object Identifier
doi:10.1215/ijm/1255985334

Mathematical Reviews number (MathSciNet)
MR1665708

Zentralblatt MATH identifier
0926.11075

#### Citation

Rivat, J.; Sárközy, A.; Stewart, C. L. Congruence properties of the $\Omega$-function on sumsets. Illinois J. Math. 43 (1999), no. 1, 1--18. doi:10.1215/ijm/1255985334. https://projecteuclid.org/euclid.ijm/1255985334