# Chen Primes in Chen's theorem

In H. Halbertsam and H. Richert's book "Sieve Methods" (Academic Press 1974) the authors go about illustrating the proof of Chen's theorem in Chapter 11 (Pages 321-337). They use a function $S_0$ defined on line (2.4) at the bottom of page 323. I simply want to know how this $S_0$ relates to the number of Chen Primes less than a given magnitude $N$. Are they the same?

-
Can you provide a hyperlink (perhaps through Google Books) to the material? Gerhard "Ask Me About System Design" Paseman, 2012.09.09 – Gerhard Paseman Sep 10 '12 at 4:51

No, they are different. This sum $S_0$ is just the contribution of numbers with three prime factors to the counting function used by Chen.
Roughly speaking, Chen first proves a lower bound for a weighted sum (say $S$, which may be different notation from Halberstam-Richert) over primes $p\leq X$ such that $p-2$ has at most three prime factors. He then gives an upper bound, which turns out to be of smaller size, to the contribution of the integers with three prime factors, and deduces that the contribution of those $p$ where $p-2$ has at most two (the "Chen primes") must be large, hence proving his theorem.