I am working from a paper by Hardy and Littlewood from 1923 which attempts to construct an approximation to the number of Goldbach pairs for a given $n$. On page 32, they present a product which explains the movement of the function they call $N_2(n)$ which gives the number of Goldbach pairs.
$$\prod_{p}\frac{p-1}{p-2}$$
where p is a distinct odd prime divsor of n.
They give a brief history and cite a few papers which I either cannot find online or are in other languages. I can intuitively understand why $N_2(n)$ would be related to $\phi(n)$, the number of totatives of n, but not the similar product these authors put forward.
My question is, is there an explanation, intuitive or otherwise, of why this product provides such a remarkable prediction of $N_2(n)$? Using their formula I get a $R^2$=99.8%.
I will try to explain with an example.
Consider one specific prime number, for example $p_i=19$.
For an even number $n$, we search how many couples $(p,q)$ of prime numbers are such that $p \le q$ and $p+q=n$
Or, how many couples $(p,q)$ of prime numbers are such that $p \le \frac{n}{2} $ and $p+q=n$ ;
We search the magnitude of 'number of couples', not the exact value, so I can exclude one specific eventual decomposition, $n=19+q$, if $n-19$ is prime.
In the range $[2, \frac{n}{2} ]$ (or in any large interval), the proportion of integers which are not multiple of $19$ is $18/19$. We can see here factor $p_i-1$
If $n$ is a multiple of $19$, when $p$ is prime (so not a multiple of $19$), we are sure that $n-p$ is not a multiple of $19$. So the proportion of couples where neither $p$ and $n-p$ is multiple of $19$ is $18/19$.
if $n$ is not a multiple of $19$, for example $n=19k+1$, if we want that neither $p$, neither $n-p$ is a multiple of $19$ , we need to eliminate all numbers $19k$ and all numbers $19k+1$, so we keep a proportion of $17/19$ only.
So, we have a bonus. Proportion $18/19$ if $n$ is a multiple of $19$, and proportion $17/19$ if $n$ is not a multiple of $19$.