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?



Upper bound on Chen primes in an interval?
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.)


