A remark on Chen's theorem. II. (Q1044788)

Let \(p\) denote a prime and \(P_2\) denote an almost prime with at most two prime factors. The author proves that for sufficiently large \(x\), \[ \pi_{1,2}(x)=\sum_{{p\leq x}\atop{p+2=P_2}} 1> \frac {1.13\,Cx}{\log^2x}, \] where the constant 1.13 constitutes an improvement of the previous result 1.104 due to \textit{J. Wu} [Acta Arith. 114, No. 3, 215--273 (2004; Zbl 1049.11107)]. The proof uses a weighted sieve method. For Part I, see Acta Arith. 102, No. 4, 339--352 (2002; Zbl 1001.11035).