On the least almost-prime in arithmetic progressions

Let \({\cal P}_{2}\) $${\mathcal{P}}_2$$ denote a positive integer with at most 2 prime factors, counted according to multiplicity. For integers a, q such that (a, q) = 1, let \({\cal P}_{2}(q, \ a)\) $${\mathcal{P}}_2(q,a)$$ denote the least \({\cal P}_{2}\) $${\mathcal{P}}_2$$ in the arithmetic progression \({\{nq+a\}_{n=1}^{\infty}}\) $$\{nq+a{\}}_{n=1}^{\mathrm{\infty }}$$ . It is proved that for sufficiently large q, we have \({\cal P}_{2}(q, \ a) \ll q^{1.825}.\) $${\mathcal{P}}_2(q,a)\ll q^{1.825}.$$

This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained \({\cal P}_{2}(q, \ a) \ll q^{1.8345}\) $${\mathcal{P}}_2(q,a)\ll q^{1.8345}$$ .