On the exceptional set of Goldbach's problem in short intervals
type: Item
instance of: scholarly article
title: On the exceptional set of Goldbach's problem in short intervals
zbMATH Open document ID: 1059.11059
published in: Monatshefte für Mathematik
publication date: 2004-09-22T00:00:00Z
review text: Let \(X\) and \(H\) be real numbers exceeding 2, and define \(E(X,H)\) to be the number of even integers in the interval \([X, X+H]\) that cannot be written as a sum of two primes. \textit{C.-H. Jia} [Acta Arith. 77, 207--287 (1996; Zbl 0863.11066)] proved that for any given positive numbers \(A\) and \(\varepsilon\), one has \(E(X,H)\ll H(\log X)^{-A}\) provided that \(H\geq X^{7/108+\varepsilon}\). But when we would assure a bound of the sharper form \(E(X,H)\ll H^{1-\delta}\) with some fixed positive number \(\delta\), currently we must admit a stronger restriction on \(H\). In this paper, the author shows that if \(\delta\) is a sufficiently small positive number and \(H\geq X^{7/24+7\delta}\), then one has \(E(X,H)\ll H^{1-\delta/600}\). This theorem improves the previously best result in the latter direction due to \textit{T. P. Peneva} [Monatsh. Math. 132, 49--65 (2001; Zbl 0974.11037); see also ibid. 141, No. 3, 209--217 (2004; Zbl 1111.11312)] in which essentially \(1/3\) took the place of the exponent \(7/24\) above. Roughly speaking, the proof is constructed by adding the idea of the localisation trick to the above work of Peneva.
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 2102089
zbMATH Keywords: Goldbach problem
zbMATH Keywords: Goldbach numbers
zbMATH Keywords: exceptional set
zbMATH Keywords: short intervals
author: Alessandro Languasco
reviewed by: Koichi Kawada
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2102089