The Hardy-Littlewood Conjecture for 3-term arithmetic progressions is that $ \# \{ x,d \in \{1,\ldots,N\} \, | \, x,x+d,x+2d \text{ are all prime} \} \sim \frac{3}{2} \prod_{p > 2} \left(1+\frac{1}{(p-1)^2}\right) \frac{N^2}{(\log N)^3}. $
In this (piece) of a paper (http://www.claymath.org/publications/Gauss_Dirichlet/green.pdf), Ben Green outlines a heuristic argument for the (k-term version) of the conjecture that I am trying to understand. I will repeat the most important parts here.
For large $N$, the probability that an arbitrary integer $\leq N$ is prime is $ \mathbb{P}(x \text{ is prime} | 1 \leq x \leq N) \approx \frac{1}{\log N} $ by the Prime Number Theorem.
Choose $x,d \in \{1,\ldots,N\}$ at random among the $N^2$ choices and write $E_j$ for the event that $x+jd$ is prime. If the events $E_0, E_1, E_2$ were independent, we would expect that $ \mathbb{P}(x,x+d,x+2d \text{ are all prime}) = \mathbb{P}(E_0 \cap E_1 \cap E_2) \approx \frac{1}{(\log N)^3}, $ and so $ \# \{x,d \in \{1,\ldots,N\} \, | \, x,x+d,x+2d \text{ are all prime} \} \approx \frac{N^2}{(\log N)^3}, $ which is the correct result up to a constant factor.
Green says that the correct constant can be obtained by discarding the incorrect assumption of independence and taking account of the fact that the primes $> q$ fall only in those residue classes $a(\text{mod }q)$ with $a$ coprime to $q$. He gives no more details.
I've been trying to figure out how to do this, but haven't been successful.
Could someone please help me out or point me to a reference where it is done?
Thanks.