On the least almost-prime in arithmetic progressions

summary:Let P₂ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers a, q such that $(a,q)=1$, let P₂(q,a) denote the least P₂ in the arithmetic progression +a\ₙ₌₁∞. It is proved that for sufficiently large q, we have P₂(q,a) q1.825. This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained P₂(q,a) q1.8345.