The differences between consecutive almost-primes (Q1103670)

Let \(d_ n=P_{2,n+1}-P_{2,n}\) denote the difference between consecutive \(P_ 2\) numbers, that are defined as having at most two prime factors, and consider the sum \(D_ 2(x)=\sum d\) \(2_ n\) summed over those n for which \(P_{2,n}\leq x\). Using analytic methods, \textit{U. Meyer} showed [Arch. Math. 42, 448-454 (1984; Zbl 0524.10036)] that \(D_ 2(x)\ll x^{1.285} \log^{10}x.\) The author shows \(D_ 2(x)\ll x^{1.023},\) using a form of the weighted sieve involving Selberg's \(\lambda\) 2-method obtained by the reviewer [Acta Arith. 47, 71-96 (1986; Zbl 0549.10036)].