Product of three primes in large arithmetic progressions

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 $(q-1,4\cdot 7\cdot 11\cdot 17\cdot 23\cdot 29)=2$, 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 $q(\log q{)}^6$.