On the existence of products of primes in arithmetic progressions

Abstract

AbstractWe study the existence of products of primes in arithmetic progressions, building on the work of Ramaré and Walker. One of our main results is that if is a large modulus, then any invertible residue class mod contains a product of three primes where each prime is at most . 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.