## Duke Mathematical Journal

### When the sieve works

#### Abstract

We are interested in classifying those sets of primes $\mathcal {P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal {P}^{c}$ we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length $x$ with primes including some in $(\sqrt{x},x]$, using methods motivated by additive combinatorics.

#### Article information

Source
Duke Math. J. Volume 164, Number 10 (2015), 1935-1969.

Dates
Revised: 9 September 2014
First available in Project Euclid: 14 July 2015

https://projecteuclid.org/euclid.dmj/1436909415

Digital Object Identifier
doi:10.1215/00127094-3120891

Mathematical Reviews number (MathSciNet)
MR3369306

Zentralblatt MATH identifier
1326.11055

Subjects
Primary: 11N35: Sieves
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity

#### Citation

Granville, Andrew; Koukoulopoulos, Dimitris; Matomäki, Kaisa. When the sieve works. Duke Math. J. 164 (2015), no. 10, 1935--1969. doi:10.1215/00127094-3120891. https://projecteuclid.org/euclid.dmj/1436909415

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