On Chen's theorem, Goldbach's conjecture and almost prime twins III

Let N denote a sufficiently large even integer and x denote a sufficiently large integer, we define D1,r(N, c, θ ) as the number of primes p such that $N - p$ has at most r prime factors and p ∈ [c N, c N+Nθ]. In this paper, we mention an important 1979 result of Lou and Yao, and use this result to give positive lower bounds for D1,3(N, 1/2, 0.872) and D1,3(N, 0, 0.817), improving on the author's previous results. Similar methods can be used to give a simplified proof for the positive lower bounds for D1,2(N, 1/2, 0.97) and D1,2(N, 0, 0.9409).