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.

arXiv comment26 pages. Final version, published in Duke Math. J. Extended the results of Section 2. Some other minor changes

References in corpus (1)

• The number of unsieved integers up to x

Cited by in corpus (1)

• Quadratic Fields Admitting Elliptic Curves with Rational $j$-Invariant and Good Reduction Everywhere