The weak Goldbach conjecture
type: Item
instance of: scholarly article
zbMATH Open document ID: 1296.11129
publication date: 2014-08-28T00:00:00Z
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 6335458
zbMATH Keywords: ternary Goldbach conjecture
zbMATH Keywords: circle method
zbMATH Keywords: Dirichlet \(L\) functions
zbMATH Keywords: minor arc
author: Harald A. Helfgott
MaRDI profile type: Publication
title: The weak Goldbach conjecture
published in: La Gaceta de la Real Sociedad Matemática Española
review text: 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.
reviewed by: Octavio Paniagua-Taboada
description: scientific article; zbMATH DE number 6335458