I have a question. My professor in the lecture said that Vinogradov's method by applying the Hardy-Littlewood circle method (minor and major arc) for the ternary Goldbach problem can be used to prove an "almost all" result for the Binary Goldbach problem. More precisely Defining $$ r(n) = \sum_{p_1 + p_2 = n} (\log p_1) (\log p_2) $$ and denoting $$ E(n) = \# \{ n \leq N: 2n \text{ is not a sum of two primes} \} $$ then By the following theorem $$ \sum_{\substack{ n=1 \\ 2 \mid n}}^{N} (r(n) - n \sigma(n) )^2 \ll N^3 (\log N)^{-A} $$ where $$ \sigma(n) = \prod_{p \mid n } \left( 1 + \frac{1}{p-1} \right) \prod_{p \nmid n} \left(1 - \frac{1}{(p-1)^2} \right) $$ it will follows that $ E(N) = O(N (\log N)^{-A} ) $
I have three question:
- Where I can find a reference of this proof?
- For the proof of Vinogradov's theorem we use $ \Lambda $ instead of $\log $ in the definition of $r(n)$, and he prove that for all but finitely many odd integers are sums of three primes by showing that the contribution of $r(n)$ made by primes power is $O( N^{3/2} \log^2 N )$. Here what is the error term that we need to showing a similar results for the binary Goldbach problem ?
- I suppose that the minor and major arc method is too weak to solve this problem, why can solve for all but finitely many integers the Ternary Goldbach problem but it can't solve a similar result for the Binary Goldbach problem?