Every odd number greater than 1 is the sum of at most five primes (Q2871196)

Goldbach's problem states that every even number \(N \geq 4\) is the sum of two primes and every odd number \(N \geq 7\) is the sum of three primes. The ternary problem was solved for sufficiently large odd \(N\) in 1937 by \textit{I. M. Vinogradow} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 15, 169--172 (1937; Zbl 0016.29101)], though it took several years to quantify what ``sufficiently large'' meant. The efforts of several mathematicians over almost half a century culminated in the result of \textit{M.-C. Liu} and \textit{T. Wang} [Acta Arith. 105, No. 2, 133--175 (2002; Zbl 1019.11026)] that Vinogradov's result holds for \(N \geq e^{3100}\). Separate work by \textit{J. M. Deshouillers} et al. [Electron. Res. Announc. Am. Math. Soc. 3, No. 15, 99--104 (1997; Zbl 0892.11032)] established the three-primes theorem for all \(N \geq 7\) under the assumption of the generalized Riemann hypothesis. The paper under review is the first of two great breakthrough that occurred in this area within just a few months. The author establishes that every odd integer \(N \geq 9\) as the sum of three primes and an even number \(\leq N_0 := 4 \times 10^{14}\). Since the binary Goldbach problem has been verified for even integers \(\leq N_0\), this establishes the result announced in the title of the paper. This result has since been eclipsed by the work of \textit{H. A. Helfgott} [``The ternary Goldbach problem'', Preprint, \url{arXiv:1501.05438}], where he finally proved the ternary Goldbach conjecture for all \(N \geq 7\).NEWLINENEWLINEThe bulk of the paper is dedicated to the study of the minor arcs in the application of the circle method to Goldbach's problem. In order to obtain an effective version of Vinogradov's theorem, one needs \(L^2\) and \(L^\infty\) bounds for the exponential sum NEWLINE\[NEWLINE S_{\eta,q}(N,\alpha) = \sum_{\gcd(n,q) = 1} \Lambda(n)\eta(n/N)\exp(2\pi i\alpha n), NEWLINE\]NEWLINE where \(q\) is a fixed modulus, \(\eta\) is a piecewise smooth weight function, \(\Lambda(n)\) is von Mangoldt's function, and \(\alpha\) is a real number that is not too close to a rational with a small denominator (\(\alpha\) is ``on a minor arc'' in circle method jargon). In the classical setting, one chooses \(q = 1\) and \(\eta\) to be the characteristic function of \([0,1]\), but those choices lead to serious losses for smallish \(N\). The author uses clever harmonic analysis to obtain sharp estimates for the above sum under quite general assumptions. Those estimates turn out to be very useful to reduce the search range from \(N \leq e^{3100}\) down to \(N \leq 8.7 \times 10^{36}\); the latter range can then be covered using effective results on primes in short intervals due to \textit{O. Ramaré} and \textit{Y. Saouter} [J. Number Theory 98, No. 1, 10--33 (2003; Zbl 1032.11038)].NEWLINENEWLINETo give the reader a taste of the more technical side of the paper, we state here the author's \(L^2\) bound for \(S_{q,\eta}\): Let \(H \geq 100\) and write \(D_H(\alpha) = \sum_{h=1}^H \exp(2\pi i\alpha)\) for the Dirichlet kernel of degree \(H\). For an appropriately chosen \(q\), one has NEWLINE\[NEWLINE \int_0^1 |S_{\eta,q}(N,\alpha)|^2|D_H(\alpha)|^2 \, d\alpha \leq (1 + \delta(H,N)) \times 8H^2N\| \eta \|_{L^\infty (\mathbb R)}^2, NEWLINE\]NEWLINE where \(\delta(H,N) = 0.13H^{-1}\log N + \varepsilon(H)\) with \(\varepsilon(H)\) an explicit function of \(H\) that decays relatively fast. In the range \(\log N \leq 3100\), the function \(\delta(H,N)\) is quite small even when \(H = 1000\), so the above inequality essentially saves a power of \(\log N\) over the classical bound NEWLINE\[NEWLINE \int_0^1 |S_{\eta_0,1}(N,\alpha)|^2 \, d\alpha \leq (1+o(1))N\log N, NEWLINE\]NEWLINE where \(\eta_0\) is the characteristic function of \((1/2, 1]\).