Closing the gap: the ternary Goldbach conjecture, unconditionally (Helfgott, 2013)

Step 2 of 5: The circle method: turning counting into an integral

In plain words

Weighting primes by the von Mangoldt function $\Lambda$ , the number of ways to write $n=p_1+p_2+p_3$ is exactly the integral of $S(\alpha)^3 e(-n\alpha)$ over the unit circle, where $S(\alpha)=\sum_p\Lambda(p)e(\alpha p)$ is a 'prime-detecting' exponential sum. This turns additive number theory into harmonic analysis: showing $R(n)>0$ reduces to understanding how $S(\alpha)$ behaves as $\alpha$ ranges over $[0,1)$ , which splits naturally into places where $S(\alpha)$ is large and structured (near rationals with small denominator — the 'major arcs') and places where it should be small and essentially random ('minor arcs').

R(n) = \sum_{p_1+p_2+p_3=n} \Lambda(p_1)\Lambda(p_2)\Lambda(p_3) = \int_0^1 S(\alpha)^3 e(-n\alpha)\,d\alpha,\qquad S(\alpha)=\sum_{p\le n}\Lambda(p)e(\alpha p)

Detailed analysis

The unit interval is dissected into major arcs $\mathfrak{M}$ (small neighborhoods of rationals $a/q$ with $q$ up to some threshold) and minor arcs $\mathfrak{m}$ (everything else). Splitting $R(n)=\int_{\mathfrak{M}} + \int_{\mathfrak{m}}$ , the strategy is to show the major-arc integral gives an explicit main term $\mathfrak{S}(n)\cdot n^2/2$ (a positive constant $\mathfrak{S}(n)$ , the 'singular series', times the expected order of magnitude), while the minor-arc integral is a provably smaller error term — which is exactly the two problems tackled in the next two steps.