On Chen's theorem. II. (Q2483167)

This paper improves the numerical constants in \textit{J. Chen}'s theorems [Sci. Sin. 16, 157--176 (1973; Zbl 0319.10056)] concerning the number of representations of a large even integer as \(p+ P_2\) and the number of primes \(p= P_2+ 2\) up to \(x\). For the Goldbach problem the constant 0.836 of \textit{J. Wu} [Acta Arith. 114, No. 3, 215--273 (2004; Zbl 1049.11107)] is replaced by 0.867, while for twin primes Wu's value of 1.104 is improved to 1.123. Moreover one can write an even integer \(N\) as \(p+ P_2\) with \(|p-N/2|\leq N^{0.97}\), improving Wu's exponent of 0.971. These results come about by combining sieve procedures from a number of earlier works.