Proposition 8.3 — strongly major arc estimate

Proposition 8.3 of Tao's five-primes paper: the strongly major arc estimate.

Fix the constants of Section 8 of the paper: the Goldbach verification height $N_0 = 4\times10^{14}$, the Bourgain scale parameter $K = 10^3$, and the Riemann-hypothesis verification height $T_0 = 3.29\times10^9$. Let $x$ be an integer with

Write $\eta_1(t) = (1 - 10\,\mathrm{dist}(t,[0.2,0.8]))_+$ and $\eta_0(t) = 4(\log 2 - |\log 2t|)_+$ for the two cutoffs, $S_{\eta,q}(y,\alpha) = \sum_n \Lambda(n)\mathbf 1_{(n,q)=1}\eta(n/y)e(n\alpha)$ for the smoothed prime exponential sum, $D_H(\alpha) = \sum_{1\le n\le H} e(n\alpha)$ for the Dirichlet kernel, and

for the circle-method integrand of equation (8.11). Set

Then the integral of $F_x$ over the strongly major arc satisfies

The paper writes this conclusion as $\frac{x^2}{K}(N_0/3)^3\bigl(\frac23 + \mathcal O^*(0.1)\bigr)$, where $\mathcal O^*(c)$ denotes a quantity of magnitude at most $c$.

This is the main term of the circle method for Theorem 8.2: it says the near-zero frequencies already contribute the full expected size $\frac23\frac{x^2}{K}(N_0/3)^3$ of the weighted representation count, with a relative error of at most $0.1$. Paired with the complementary minor arc bound it yields the positivity of that count, which is the one remaining analytic obligation of the whole five-primes argument.

Formalization Note The circle is AddCircle 1 carrying its Haar probability measure, and $\|\alpha\|_{\mathbb R/\mathbb Z}$ is its norm, the distance from $\alpha$ to the nearest integer; the strongly major arc is therefore the closed ball of radius $T_0/(3.6\pi x)$ about $0$. The integrand is the published representationIntegrand, whose three finite Fourier polynomials agree with $S_{\eta_1,\sqrt x\sharp}(x,\cdot)$, $S_{\eta_0,\sqrt{x/K}\sharp}(x/K,\cdot)$ and $D_{N_0/3}$ because $\eta_1$ is supported in $[0.1,0.9]$ and $\eta_0$ in $(1/4,1)$, so the truncated ranges lose nothing.

The Dirichlet kernel sums over $1 \le n \le \lfloor N_0/3\rfloor$, which is what the paper's $\sum_{1\le n\le N_0/3}$ means, and $M$ is written with $\lfloor N_0/3\rfloor^3$ to match it; the gap between $(N_0/3)^3$ and $\lfloor N_0/3\rfloor^3$ is a relative $8\times10^{-15}$, far inside the stated slack of $0.1$.

The radius uses $3.6\pi$, as in the statement of Proposition 8.3 and in Proposition 8.4; two displays inside the proof print $3\pi$ instead, which is inconsistent with the surrounding argument. The upper bound $x \le e^{3100}$ is not needed for this proposition, whose proof uses only $x \ge 8.7\times10^{36}$; it is carried because it is a standing hypothesis of Section 8.