On the least almost-prime in arithmetic progression

Abstract

summary: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 $\mathcal {P}_2\equiv a\pmod q$. It is proved that for sufficiently large $q$, there holds $$\mathcal {P}_2(a,q)\ll q^{1.8345}.$$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$