On the least almost-prime in arithmetic progression

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 ${\mathcal{P}}_2\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)$. 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 $1.8345$.