Product of three primes in large arithmetic progressions

TL;DR

This paper proves that for large moduli, there are numbers congruent to any invertible residue class that are products of three or four primes within specific bounds, extending prime distribution in arithmetic progressions.

Contribution

It establishes new bounds for the smallest prime factors in products of three or four primes in large arithmetic progressions, generalizing previous results.

Findings

Existence of numbers as products of three primes in large progressions within bounds q^{3/2+ε}.

Improved bounds for primes in special residue classes, down to q^{11/8+ε} and q^{6/5+ε}.

Existence of four-prime products below q(log q)^6 for sufficiently large q.

Abstract

For any $\epsilon>0$, there exists $q_0(\epsilon)$ such for any $q\ge q_0(\epsilon)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+\epsilon}$. If we restrict our attention to odd moduli $q$ that do not have prime factors congruent to 1 mod 4, we can find such primes below $q^{\frac{11}{8}+\epsilon}$. If we further restrict our set of moduli to prime $q$ that are such that $(q-1,4\cdot7\cdot11\cdot17\cdot23\cdot29)=2$, we can find such primes below $q^{\frac{6}{5}+\epsilon}$. Finally, for any $\epsilon>0$, there exists $q_0(\epsilon)$ such that when $q\ge q_0(\epsilon)$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly four primes, all of which are below $q(\log q)^6$.