Let $\mathcal{ P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{ P}_2(a,q)$ the least almost-prime $\mathcal{ P}_2$ which satisfies $\mathcal{ P}_2\equiv a\pmod q$. In this paper, it is proved that for sufficiently large $q$, there holds \begin{ equation*} \mathcal{ P}_2(a,q)\ll q^{ 1.8345}. \end{ equation*} This result constitutes an improvement upon that of Iwaniec, who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$.
Categories: math.NT