Weighted sieves and twin prime type equations
type: Item
instance of: scholarly article
title: Weighted sieves and twin prime type equations
zbMATH Open document ID: 0683.10038
DOI: 10.1215/S0012-7094-89-05834-1
published in: Duke Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: By the methods of \textit{J. Chen} [Sci. Sin., Ser. A 16, 157-176 (1973; Zbl 0319.10056)] it is known that the equation \(p+2=P_ 2\) has infinitely many solutions, where p is prime and \(P_ 2\) has at most two prime factors. Chen's methods involve the use of a ``switching principle'' whereby an upper bound is obtained (and used) for the number of solutions of a subsidiary equation \(p+2=p_ 1p_ 2p_ 3\) over appropriate ranges of the primes involved. The authors give a proof of Chen's result, in a quantitatively weaker form, without using the ``switching principle''. Indeed, their methods would deal, with less difficulty, with the analogous equation \[ p_ 1... p_ k+2=P_ 2 \] for which no ``switching'' argument is available. They use a weighted sieve in a form provided by \textit{M. Laborde} [Mathematika 26, 250-257 (1979; Zbl 0429.10028)] and the result of \textit{E. Bombieri}, \textit{J. B. Friedlander} and \textit{H. Iwaniec} [Acta Math. 156, 203-251 (1986; Zbl 0588.10042)] concerning expressions of the form \(\sum_{q\leq x^ c}\lambda (q) E(x,q,a)\) where \(c<4/7\), E is the error term in the Prime Number Theorem for arithmetic progressions and \(\lambda\) (q) is a ``well-factorable'' coefficient. Since the weighted sieve leads to coefficients which are not exactly ``well-factorable'' an extension of this result has to be developed.
reviewed by: George Greaves
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 4118460
zbMATH Keywords: Goldbach-type equations
zbMATH Keywords: twin prime type equations
zbMATH Keywords: Chen's result
zbMATH Keywords: weighted sieve
author: Étienne Fouvry
author: F. Grupp
MaRDI profile type: Publication
cites work: Primes in arithmetic progressions to large moduli
cites work: Primes in arithmetic progressions to large moduli. II
cites work: Q4081355
cites work: Kloosterman sums and Fourier coefficients of cusp forms
cites work: Répartition des suites dans les progressions arithmétiques
cites work: Around the Bombieri-Vinogradov theorem
cites work: Autour du théorème de Bombieri-Vinogradov. II
cites work: Q3715206
cites work: Primes in arithmetic progressions
cites work: Incomplete Kloosterman sums and a divisor problem. Appendix: On some exponential sums by Bryan J. Birch and Enrico Bombieri
cites work: A weighted sieve of Brun's type
cites work: Q3720678
cites work: Q3901584
cites work: Q4052222
cites work: Q3720679
cites work: Q3720680
cites work: ARTIN'S CONJECTURE FOR PRIMITIVE ROOTS
cites work: The divisor function $d_3(n)$ in arithmetic progressions
cites work: A new form of the error term in the linear sieve
cites work: Buchstab's sifting weights
description: scientific article; zbMATH DE number 4118460