## Illinois Journal of Mathematics

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

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

J. Rivat, A. Sárközy, and C. L. Stewart

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/ijm/1255985334

**Mathematical Reviews number (MathSciNet)**

MR1665708

**Zentralblatt MATH identifier**

0926.11075

**Subjects**

Primary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions

Secondary: 11N36: Applications of sieve methods

#### 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