A source I am reading refers to the Goldbach conjecture (that every even number is the sum of two primes), and then immediately follows with the "Hardy-Littlewood conjecture" that
$\sum \limits_{n \leq N} \Lambda(n) \Lambda(N-n) = 2 \prod \limits_{p \geq 3} \left(1-\frac{1}{(p-1)^2}\right) \left( \prod \limits_{p | N} \frac{p-1}{p-2}\right)N^{1+o(1)}$
which is termed "Goldbach for almost all even numbers", and apparently this also
$\Longleftrightarrow \prod \limits_{p | N} (\frac{p-1}{p-2}) N^{1+o(1)} = O(\log N)^c$. Here, $\Lambda(n)$ denotes the Von-Mangoldt function.
It then states the following theorem:
Let $A \in \mathbb{R}$. Then for all but $\frac{x}{\log^A x}$ even numbers $N \leq x$, we have $\sum \limits_{n \leq N} \Lambda(n) \Lambda(N-n) = 2 \prod \limits_{p \geq 3} \left(1-\frac{1}{(p-1)^2}\right) \left( \prod \limits_{p | N} \frac{p-1}{p-2}\right)N (1+o_A(1))$, where $o_A$ denotes the fact that the constant in the limiting behaviour may depend on $A$.
Now I can't see how this is the same as Goldbach at all really. I can see that if $N= p+q$ is the sum of 2 primes then the corresponding term on the LHS will be nonzero, but the Von-Mangoldt function is also nonzero for powers of primes, so it might be nonzero for some $N= P^i + Q^j$. I am beginning to think the RHS may be some sort of probabilistic slant on the conjecture, but can't quite see it.
It may be the case that the Von-Mangoldt function is just negligible on all prime powers except the primes themselves (this is certainly often the case), but at the least it seems like this should have been stated somewhere since it certainly doesn't seem like a trivial deduction to me. Or is this simply an "approximation to Goldbach"; namely a relationship which seems to imply that Goldbach might well be true, but as I have said doesn't remove the problem of the prime powers? (Obviously I am aware that Goldbach is unproved, but the text doesn't clarify whether this is a proved statement weaker than Goldbach, or an unproved statement equivalent to Goldbach: I suspect the latter.)
I also can't see how the "$\Longleftrightarrow$" follows from the first conjecture; I'd be very grateful if someone could help me understand what's going on here, at least heuristically if not formally.
Next, the text goes on to "prove" Vinogradov from the latter theorem which has been stated (I say "prove" because I can't see how the proof works). It says:
Corollary (Vinogradov) Every sufficiently large odd number is the sum of 3 primes. Proof: Let N be odd. Then taking $A=2$ in the theorem, there is some prime $p\leq N/2$ for which the Hardy-Littlewood asymptotic holds. In particular, $N-p$ is the sum of 2 primes.
Now this time I really can't see what's happening: the sum over the Von-Mangoldt function on the LHS will surely just go to zero almost every time $N-n$ is a sum of 2 primes (unless this sum of 2 primes is a prime power of course), and I don't see how the RHS tells us nothing about the LHS except that it is not "extremely small". Could anyone explain what's going on here to me?
It is possible I transcribed some of this material wrong, though I do not see where I might have made an error which could have caused all my confusions simultaneously. Again, any insight you could provide would be desperately appreciated; many thanks in advance.