Explicit bounds for products of primes in AP

Abstract

For all q ≥ 2 q\ge 2 and for all invertible residue classes a a modulo  q q , there exists a natural number that is congruent to a a modulo  q q and that is the product of exactly three primes, all of which are below ( 10 15 q ) 5 / 2 (10^{15}q)^{5/2} .