Proposition 8.4: the $L^2$ estimate off the strongly major arc

$L^2$ estimate (Proposition 8.4). Let $8.7\times10^{36}\le x\le e^{3100}$ be an integer and $N_0=4\times10^{14}$, $T_0=3.29\times10^9$. Then

(The paper states the bound for $\|\alpha\|\ge T_0/(3.6\pi x)$; the boundary has measure zero, and the version over the open complement of majorArc x is what the assembly consumes.)

In the paper this follows from Proposition 4.10 — the large sieve with Siebert's explicit bound on prime pairs, which gives $\int_{\mathbb R/\mathbb Z}|S_{\eta_1,\sqrt x\sharp}|^2|D_{N_0/3}|^2\le 8H^2x\big(1+\frac{0.13\log x}{H}+\frac{(e^\gamma\log\log(2H)+\frac{2.507}{\log\log(2H)})\log(9H)}{2H}\big)$ with $H=\lfloor N_0/3\rfloor$, hence $\le 8.001\,(N_0/3)^2x$ using $\log x\le3100$ — and a lower bound $\ge0.919\,(N_0/3)^2x$ for the strongly major arc contribution from Corollary 4.8 (with $c=0.1$ and the norms (8.3), (8.7), (8.9)) and (8.13). Subtracting gives $7.09$. The point of using the large sieve rather than Plancherel is that it loses only a constant factor ($8$), not a factor of $\log x$.

Retired. The constant 7.09 rests on the major-arc lower bound 0.92x on p. 41, which appears to be an error (Corollary 4.9 is normalised to ‖η‖₂ = 1 while ‖η₁‖₂² = 2/3; the correct lower bound is ≈ 0.627x, so the difference is ≈ 7.37). Replaced by TaoFivePrimes.minor_arc_L2_corrected (constant 7.4). The final argument is unaffected because the companion L^∞ bound holds with 0.05 (see minor_arc_Linfty_corrected).