Some remarks on Goldbach's problem
type: Item
instance of: scholarly article
title: Some remarks on Goldbach's problem
zbMATH Open document ID: 0808.11060
DOI: 10.1007/BF01305775
author: Gunter Dufner
author: Dieter Wolke
published in: Monatshefte für Mathematik
publication date: 1994-10-09T00:00:00Z
full work available at URL: https://eudml.org/doc/178658
review text: It is well-known that there exist sparse sets of integers \({\mathcal A}\) such that the equation \(n=a+p\) is soluble for all sufficiently large \(n\), where \(a\in{\mathcal A}\) and \(p\in{\mathcal P}\), the set of all prime numbers. Refining combinatorial methods due to \textit{G. G. Lorentz} [Proc. Am. Math. Soc. 5, 838-841 (1954; Zbl 0056.039)] and \textit{P. Erdős} [Proc. Am. Math. Soc. 5, 847-853 (1954; Zbl 0056.270)], the authors prove that for all \(A>0\) there exists a set \({\mathcal P}'\subseteq {\mathcal P}\) such that \[ |\{p\leq x,\;p\in{\mathcal P}'\} |\ll_ A (\log x)^ 2, \] for all even numbers \(n\leq x\) with at most \(O(x(\log x)^{-A})\) exceptions, the equation \(n=p+ p'\) can be solved with \(p\in{\mathcal P}\) and \(p'\in{\mathcal P}'\), and for all sufficiently large odd \(n\) the equation \(n=p_ 1+ p_ 2+ p'\) can be solved with \(p_ 1\), \(p_ 2\in{\mathcal P}\) and \(p'\in{\mathcal P}'\). Following, \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arith. 27, 353-370 (1975; Zbl 0301.10043)], the exceptional set is improved to \(O(x^{1- \delta})\) for some \(\delta>0\), at the cost of having a larger set \({\mathcal P}'\), satisfying \[ |\{p\leq x,\;p\in{\mathcal P}'\}| \ll_ \varepsilon x^ \varepsilon. \] The ternary result is also improved by showing the existence of a set \({\mathcal P}' \subseteq{\mathcal P}\) such that \[ |\{p\leq x,\;p\in{\mathcal P}'\}| \ll\log \log x. \] The proofs depend on lower bounds for the number of Goldbach representations of almost all even numbers in short intervals.
reviewed by: Alessandro Zaccagnini
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 650850
zbMATH Keywords: binary Goldbach problem
zbMATH Keywords: ternary Goldbach problem
zbMATH Keywords: exceptional set
zbMATH Keywords: lower bounds
zbMATH Keywords: number of Goldbach representations of almost all even numbers
zbMATH Keywords: short intervals
MaRDI profile type: Publication
cites work: Q3658076
cites work: Sur l'hypothèse de Goldbach pour presque tous les nombres pairs
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OpenAlex ID: W2329202443
description: scientific article; zbMATH DE number 650850