Heuristic Proof of Hardy-Littlewood Conjecture for 3-term Arithmetic Progressions 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.

There are many conjectures in Number Theory with a constant factor given as that kind of product over primes, obtained by diddling with independence conjectures. You will find some of them, with explanations, by searching for Bateman Horn, and what you find should be applicable to the arithmetic progressions problem. 
