On the existence of products of primes in arithmetic progressions

TL;DR

This paper proves that for large moduli, every invertible residue class contains a product of three small primes, using advanced techniques from sieve theory, additive combinatorics, and character sum estimates.

Contribution

It introduces a novel approach combining Heath-Brown's character sum results with sieve and additive methods to establish the existence of such prime products in arithmetic progressions.

Findings

Every invertible residue class mod q contains a product of three primes up to q^{6/5+ε} for large q.

The method integrates Heath-Brown's character sum bounds into the study of prime products.

The results extend previous work by providing explicit bounds and new techniques.

Abstract

We study the existence of products of primes in arithmetic progressions, building on the work of Ramar\'e and Walker. One of our main results is that if $q$ is a large modulus, then any invertible residue class mod $q$ contains a product of three primes where each prime is at most $q^{6/5+\epsilon}$. Our arguments use results from a wide range of areas, such as sieve theory or additive combinatorics, and one of our key ingredients, which has not been used in this setting before, is a result by Heath-Brown on character sums over primes from his paper on Linnik's theorem.