The weak Goldbach conjecture
In this article the author explains his solution of the ternary (also called weak) Goldbach conjecture and the history behind this old problem. It appeared for the first time in the correspondence between Euler and Goldbach. In modern terms it states that any odd integer greater that 5 can be written as the sum of three prime numbers.NEWLINENEWLINEThe fist big step toward the conjecture was given by \textit{I. M. Vinogradov} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 15, 169--172 (1937; Zbl 0016.29101)], proving the conjecture for all odd integers bigger than some unspecified constant \(C\). This was already proven by \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Acta Math. 41, 119--196 (1917; JFM 46.0498.01)] but under the assumption of the Generalized Riemann Hypothesis. Since then the constant \(C\) has been specified and improved gradually. The lowest value known was \(C= e^{3100}\), see [\textit{M.-C. Liu} and \textit{T. Wang}, Acta Arith. 105, No. 2, 133--175 (2002; Zbl 1019.11026)]. The author reduced this constant to \(C= 10^{29}\) [``Major arcs for Goldbach's problem'', Preprint, \url{arXiv:1305.2897}]. With this bound he achieved the proof of the ternary Goldbach conjecture. NEWLINENEWLINENEWLINE The main ingredients in the author's proof, like the circle method due to Hardy-Littlewood, Dirichlet \(L\)-functions and their poles, and major arcs estimations are also explained. The article is written in a very comprehensible language.
scientific article; zbMATH DE number 6335458