Products of primes in arithmetic progressions: a footnote in parity breaking

TL;DR

This paper proves that for certain parameters, any invertible residue class modulo q can be represented by a product of exactly three small primes, extending understanding of prime products in arithmetic progressions.

Contribution

It establishes the existence of a product of three small primes in each invertible residue class modulo q under specific size constraints, advancing results in prime distribution in arithmetic progressions.

Findings

Existence of a product of three primes in each invertible residue class modulo q

Primes involved are all below x^{1/3}

Applicable for q up to x^{1/16}

Abstract

We prove that, if $x$ and $q\leqslant x^{1/16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1/3}$, that is congruent to $a$ modulo $q$.