On the A-dependence of the linear sieve with application to polynomial sequences
type: Item
instance of: scholarly article
title: On the A-dependence of the linear sieve with application to polynomial sequences
zbMATH Open document ID: 0603.10045
DOI: 10.1016/0022-314X(86)90098-3
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: This paper is concerned with error terms in the linear sieve. In the notation of \textit{H. Halberstam} and \textit{H.-E. Richert}'s book ''Sieve Methods'' (1974; Zbl 0298.10026), the linear sieve gives the bounds \[ S(A,P,z)\leq XV(z) \{F(u)+H\}+\sum_{d<y,\;d | P(z)}| R_ d|, \] \[ S(A,P,z)\geq XV(z) \{f(u)-H\}+\sum_{d<y,\;d | P(z)}| R_ d|. \] Here, \(H\) is an error term which depends on the constant \(A_ 2\) in the hypothesis \[ V(v)/V(w)\leq (\log v/\log w)(1+A_ 2/\log v),\;2\leq v\leq w. \] (This is hypothesis \(\Omega_ 2(1,A_ 2)\) of ''Sieve methods''.) Halberstam and Richert proved the estimate \(H\ll_{A_ 2} (\log X)^{-1/14}.\) \textit{H. Iwaniec} [Acta Arith. 36, 171--202 (1980; Zbl 0435.10029)] was the first to give an estimate for \(H\) with the dependence on \(A_ 2\) made explicit; he proved that \[ H=(\log y)^{-1/3}\;\exp (A_ 2^{1/2}-u \log u+u \log \log 3u+O(u)). \] Richert (unpublished) improved this to \(H\ll A_ 2^ 8 e^{-u}(\log z)^{-1/3}\). In this paper, the author replaces \(\Omega_ 2(1,A_ 2)\) with \((\Omega_ a)\): There exists a constant \(A\geq 2\) such that \[ V(v)/V(w)\leq (\log v/\log w)(1+A/\log v)\quad\text{if}\;a\leq v<w. \] With this assumption, he proves that \[ H \ll_ a V^{-1} A^{4+\alpha} (\log A)^{2+\alpha} (\log z)^{-\alpha}\quad \text{for}\;0<\alpha \leq 1/3. \] As an application, the author gives upper bounds on the number of primes in polynomial sequences. Unlike previous bounds of this sort, his bounds are uniform in the discriminant of the corresponding polynomial.
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11N13
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 3972981
zbMATH Keywords: error terms
zbMATH Keywords: linear sieve
zbMATH Keywords: upper bounds
zbMATH Keywords: number of primes in polynomial sequences
author: Gerhard Bantle
reviewed by: Sidney W. Graham
MaRDI profile type: Publication
cites work: Q3739229
cites work: Q4052222
cites work: Q3720679
cites work: Rosser's sieve
cites work: A new form of the error term in the linear sieve
cites work: An improvement of Selberg's sieve method I
cites work: On the density of some sets of primes, II
full work available at URL: 0022-314x(86)90098-3
OpenAlex ID: W2062253456
Recommended article: One-sided sifting density hypotheses in Selberg's sieve
Recommended article: Q3715208
Recommended article: Q2723209
Recommended article: Q4309281
Recommended article: Q3204561
Recommended article: Q4249512
Recommended article: Q3577570
Recommended article: Q4731256
Recommended article: The weighted linear sieve and Selberg's $λ^2$-method
Recommended article: Lectures on sieve methods and prime number theory
description: scientific article; zbMATH DE number 3972981