Upper bound on Chen primes in an interval? I'm well aware of the fact that the number of Chen primes between $N/2$ and $N$ for large enough $N$ is at least 
$$\frac{c_1N}{\ln^2(N)}$$
(Green and Tao).  My question is: is there possibly an upper bound for chen primes between $N/2$ and $N$?  What I am eventually trying to prove is that there are infinitely many intervals 
$$\Omega_i=[i^2/2, i^2]$$
such that the number of chen primes in $\Omega_i$ is at least 
$$\frac{c_1i^2}{13\ln(i^2)}.$$
Does anyone know where to look?
 A: Yes. Sieve methods are much better at proving upper bounds than lower bounds. By the Selberg sieve, or alternatively using the combinatorial sieve, you can prove that the number of Chen primes is bounded above by $C_2 N/\log^2 N$ for some particular value of $C_2$. (ed: Please see Terry Tao's important caveat below, which I neglected in my initial answer.)
I don't know the details offhand; it would take some work to determine a particular value of $C_2$, but this is definitely possible. The Selberg sieve is probably easiest, you can read, for example, Halberstam and Richert. (Just read the first chapter on the Selberg sieve -- no need to delve into the more difficult portions of the book.)
A: Explicit lower bounds for twin almost primes up to $x$ (for large enough $x$) are known, see e.g., Corollary 25.12 in the book of Friedlander-Iwaniec. (The constant they get for $p$ with $p+2$ having at most two prime factors, each of which is at least $x^{3/11}$, is $1/31$). I don't think the known upper bounds are good enough to deduce the corresponding statement between $x/2$ and $x$. Since moreover the use of primes to start the linear sieve blocks it from being "local" enough, I don't think we know lower bounds for this question for all dyadic intervals (in other words, there does not seem to exist a version of, say, the Bombieri-Vinogradov Theorem for a dyadic interval). But "almost all" such intervals might possibly be accessible differently.
By the way, I'm curious as to why a result of Chen seems to be attributed to Green-Tao in the original question...
