Vinogradov's theorem

Remarks

Vinogradov proved the theorem in 1937 by the Hardy–Littlewood circle method. The proof gives an asymptotic formula: for every $A > 0$, $$\sum_{p_1 + p_2 + p_3 = N} \log p_1 \log p_2 \log p_3 = \tfrac12 \mathfrak{S}(N) N^2 + O\!\left(\frac{N^2}{(\log N)^{A}}\right),$$ where $$\mathfrak{S}(N) = \prod_{p \mid N} \left(1 - \frac{1}{(p-1)^2}\right) \prod_{p \nmid N} \left(1 + \frac{1}{(p-1)^3}\right).$$ For odd $N$ the singular series satisfies $\mathfrak{S}(N) > 0.6$, so the main term dominates for large $N$; for even $N$ the factor at $p = 2$ is $0$. The major arcs are handled by the Siegel–Walfisz theorem, and the new step was Vinogradov's bound for the exponential sum $\sum_{p \le N} \log p \; e(p\alpha)$ on the minor arcs. The first proof gave no explicit $N_0$, because the Siegel–Walfisz constants are ineffective; later proofs made $N_0$ explicit. In 2013 Harald Helfgott proved that every odd integer greater than $5$ is a sum of three primes.

Going deeper

For a student

The theorem is the model application of the circle method to primes. The case of three primes works because the minor arcs can be bounded with one factor estimated pointwise and the other two in mean square, using Parseval's identity. For two primes, which is Goldbach's conjecture, this argument fails: the mean square alone is as large as the expected main term. The strongest results towards Goldbach's conjecture by other means are sieve results such as Chen's theorem.