On the exceptional set in Goldbach's problem in short intervals
type: Item
instance of: scholarly article
title: On the exceptional set in Goldbach's problem in short intervals
zbMATH Open document ID: 1167.11035
author: Kaisa Matomäki
published in: Monatshefte für Mathematik
publication date: 2008-11-25T00:00:00Z
review text: Let \(E(X,H)\) denote the number of even integers \(n\) with \(X<n\leq X+H\) such that \(n\) cannot be written as a sum of two primes, and set \(H=X^{\theta}\). It is known that \(E(X,H)=o(H)\) as \(X\rightarrow\infty\), if \(\theta\) is larger than a certain small number. In fact, \textit{G. Harman} showed that if \(\theta\geq11/180\), then for any fixed \(A\), one has \(E(X,H)\ll H(\log X)^{-A}\) (see Chapter 10 of [Prime-detecting sieves. London Mathematical Society Monographs 33. Princeton, NJ: Princeton University Press (2007; Zbl 1220.11118)]). Meanwhile, the methods of H. L. Montgomery and R. C. Vaughan enabled us to establish a sharper bound of the form \(E(X,H)\ll H^{1-\delta}\) with some constant \(\delta>0\), at least when \(H\) is as large as \(X\). Several mathematicians have worked to show bounds of the latter type for smaller \(H\), and for example \textit{T. P. Peneva} [Monatsh. Math. 132, No. 1, 49--65 (2001; Zbl 0974.11037), Corrigendum 141, No. 3, 209--217 (2004; Zbl 1111.11312)] and \textit{A. Languasco} [Monatsh. Math. 141, No. 2, 147--169 (2004; Zbl 1059.11059)] showed such bounds, respectively, for \(\theta>1/3\) and \(\theta>7/24\). And in the paper under review, the author proves that for \(\theta>1/5\), one has \(E(X,H)\ll H^{1-\delta}\) with some effectively computable positive number \(\delta\) depending on \(\theta\), substantially improving previous results in this direction. (The author points out, and overcomes, a flaw in the argument of Languasco (loc. cit.).) It may be said in brief that the proof is constructed by incorporating ideas of sieve methods into the previous work being based on the circle method.
reviewed by: Koichi Kawada
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 5373510
zbMATH Keywords: Goldbach problem
zbMATH Keywords: circle method
zbMATH Keywords: sieve methods
MaRDI profile type: Publication
full work available at URL: s00605-008-0561-1
OpenAlex ID: W2040535243
cites work: Q4250381
cites work: Q4850744
cites work: A large sieve density estimate near \(\sigma = 1\)
cites work: On the Distribution of αp; Modulo One
cites work: On the Distribution of αp Modulo One II
cites work: Q5754222
cites work: Q4893761
cites work: Q4830109
cites work: On the exceptional set of Goldbach numbers in a short interval
cites work: On the exceptional set of Goldbach's problem in short intervals
cites work: The exceptional set of Goldbach numbers (II)
cites work: Q3926692
cites work: The exceptional set of Goldbach's problem
cites work: On the exceptional set for Goldbach's problem in short intervals
cites work: Corrigendum: ``On the exceptional set for Goldbach's problem in short intervals''
cites work: On the Exceptional Set for Goldbach's Problem in Short Intervals
cites work: Q4175321
cites work: On the fractional parts of {x/n} and related sequences. II, III
cites work: Q3957034
DOI: 10.1007/S00605-008-0561-1
description: scientific article; zbMATH DE number 5373510