On the least almost-prime in arithmetic progression

Abstract

Let ${{\cal P}_r}$ 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 ${{\cal P}_2}\left({a,q} \right)$ the least almost-prime ${{\cal P}_2}$ which satisfies ${{\cal P}_2} \equiv a$ (mod 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.