On the existence of products of primes in arithmetic progressions.

We 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 q$q$ is a large modulus, then any invertible residue class mod q$q$ contains a product of three primes where each prime is at most q6/5+ε$q^{6/5+\epsilon }$. 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. [ABSTRACT FROM AUTHOR]