Products of primes in arithmetic progressions

Erdös conjectured that, when $q$ is a sufficiently large prime, every residue class $\pmod{q}$ can be represented as a product of two primes $p_1p_2$ with $p_1, p_2 \leq q$. This can be seen as a multiplicative analogue of the Goldbach conjecture claiming that every even integer greater than two can be written as a sum of two primes.
I will discuss my on-going work with Joni Teräväinen establishing among other things a ternary variant of Erdös' conjecture that, for every sufficiently large cube-free $q$, every reduced residue class $\pmod{q}$ can be represented as a product of three primes $p_1 p_2 p_3$ with $p_1, p_2, p_3 \leq q$. This improves on very recent works of Szabo and Zhao showing that one has such presentations with products of six primes.
In the first part of the talk I will give a general overview of the topic as well as discuss some very fundamental ideas in the proof, in particular why the problem is more difficult than the ternary Goldbach problem and how we overcome this difficulty. In the second part of the talk I will give a more detailled description of the proof ideas.