Strategy of Helfgott when proving Weak Goldbach Conjecture

Question

In 2013, Harald Helfgott proved that the weak Goldbach conjecture (now a theorem) which states that every odd number grater than $5$ is sum of three primes.

My question: What was his strategy when he is proving the statement above?

Answer 1

It's the classical strategy of Vinogradov. Note that the number of representations of $n$ as a sum of three primes is given by $$ \int_{0}^{1} S(\alpha)^3 e(- n \alpha) d \alpha $$ where $e(x) = e^{2\pi i x}$ and $$ S(\alpha) = \sum_{p \leq X} e(p \alpha) $$ for some $X \in [2n, 4n]$, with the sum above being over prime numbers. Then the strategy is that you will analyze asymptotically the integral above. The crucial new ingredient is to keep careful track of how large $n$ needs to be for the asymptotic to kick in. The integral itself is analyzed by by splitting the $\alpha$'s into two sets known as the major and minor arcs.

First recall that by Dirichlet's theorem for all $\alpha$ and any $Q$ there is a $q \leq Q$ such that, $$ \Big | \alpha - \frac{a}{q} \Big | \leq \frac{1}{q Q}. $$ Pick $Q = X$. The set of major arcs is the set of those $\alpha$ for which there is a $q \leq \log^{A} X$ with, $$ \Big | \alpha - \frac{a}{q} \Big | \leq \frac{\log^A X}{X} $$ for some large exponent $A$. The set of minor arcs is the set of all remaining $\alpha$'s. The choice of the major arcs is dictated by limitations of the Siegel-Walfisz theorem (but one caveat is that Harald is not allowed to use this theorem as it gives no effective results, that makes his major arcs much more narrow and hence more difficult to treat well numerically).

The set of minor arcs is bounded by bounding the integral as, $$ \sup_{\alpha \in \mathfrak{m}} |S(\alpha)| \int_{0}^{1} |S(\alpha)|^2 d \alpha $$ where $\mathfrak{m}$ is the set of minor arcs. The integral over $\alpha$ is easily evaluated by Plancherel and is bounded by $N \log N$. So one needs to obtain a bound of the form, $$ \sup_{\alpha \in \mathfrak{m}} |S(\alpha)| \ll N (\log N)^{-A} $$ for some large exponent $A$. For this one needs Vinogradov's method of bilinear sums.

Now Harald goes through this quite carefully --- there are several caveats, first he is not allowed to use Siegel-Walfisz, secondly in the use of Vinogradov's method there is a loss of several logs, this is bad for numerical results, and so one needs to circumvent this loss. All in all the strategy is well-known but the tour de force is that he has more limitations to fight against and yet manages to squeeze a good constant of $n > 10^{30}$ after which the asymptotic quicks in . The remaining integers are handled by a machine computation.