Montgomery's weighted sieve for dimension two
type: Item
instance of: scholarly article
title: Montgomery's weighted sieve for dimension two
zbMATH Open document ID: 0344.10024
DOI: 10.1007/BF01540603
author: K. Appert
published in: Monatshefte für Mathematik
publication date: 1976-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/177790
review text: Montgomery's weighted sieve (see [\textit{H. L. Montgomery}, Topics in multiplicative number theory, Lect. Notes Math. 227. Berlin: Springer (1971; Zbl 0216.03501); \textit{H. L. Montgomery} and \textit{R. C. Vaughan}, Mathematika 20, 119--134 (1973; Zbl 0296.10023)]) is used to prove the following theorem: let \(a\neq 0, b\neq 0\)be integers with \((a,b)=1\), \(2\mid ab\). Then we have for \(x>1\) \[ \sum_{_{\substack{ p\leq x\\ ap+b\in\mathcal P}}} 1\leq 16\omega \frac x{\log^2x}\prod_{_{\substack{ p\mid ab\\ p>2}}} \frac{p-1}{p-2},\quad \omega:=\prod_{p>2} (1-(p-1)^{-2}), \] (\(\mathcal P\) denotes the set of all prime numbers). Recently, \textit{R. C. Vaughan} [J. Reine Angew. Math. 290, 93--108 (1977; Zbl 0344.10028)] used an estimate like this to prove that every even number \(n\geq 2\) is the sum of at most 26 primes.
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 3536207
zbMATH Keywords: Montgomery's weighted sieve, two-dimensional sieve problems
zbMATH Keywords: bounds for the number of twin primes
zbMATH Keywords: number of Goldbach representations
MaRDI profile type: Publication
cites work: Q4052222
cites work: Q4079683
cites work: Topics in multiplicative number theory
cites work: The large sieve
cites work: Approximate formulas for some functions of prime numbers
OpenAlex ID: W2070660420
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description: scientific article; zbMATH DE number 3536207