Chen's theorem with small primes. (Q1862920)

Let \(S(n,\theta)\) be the number of ways of writing an integer \(N\) in the form \(p+ P_2\), where \(p\leq N^\theta\) is prime, and \(P_2\) has at most 2 prime factors. Then it is shown that \(S(N,\theta)\ll C(N) N^\theta(\log N)^{-2}\) if \(\theta= 0.95\), where \(C(N)\) is the usual function arising from the Hardy-Littlewood singular series. The proof follows Chen's argument, using a short interval form of the generalized Bombieri-Vinogradov Theorem, due to \textit{J. Wu} [Q. J. Math., Oxf. II. Ser. 44, 109--128 (1993; Zbl 0801.11038)].