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When the sieve works II

  • Kaisa Matomäki EMAIL logo and Xuancheng Shao
Published/Copyright: December 16, 2018

Abstract

For a set of primes 𝒫, let Ψ(x;𝒫) be the number of positive integers nx all of whose prime factors lie in 𝒫. In this paper we classify the sets of primes 𝒫 such that Ψ(x;𝒫) is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki [A. Granville, D. Koukoulopoulos and K. Matomäki, When the sieve works, Duke Math. J. 164 2015, 10, 1935–1969] and their main conjecture is proved in this paper. In particular, our main theorem implies that, if not too many large primes are sieved out in the sense that

p𝒫x1/v<px1/u1p1+εu,

for some ε>0 and vu1, then

Ψ(x;𝒫)ε,vxpxp𝒫(1-1p).

Funding source: Suomen Akatemia

Award Identifier / Grant number: 137883

Award Identifier / Grant number: 138522

Funding statement: Kaisa Matomäki was supported by Academy of Finland grants no. 137883 and 138522. Xuancheng Shao was supported by a Glasstone Research Fellowship.

Acknowledgements

The present work started when both authors were visiting CRM in Montreal during the analytic part of the thematic year in number theory in Fall 2014, whose hospitality is greatly appreciated. Thanks to Ben Green for helpful discussions. Thanks to the anonymous referee for valuable comments and suggestions.

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Received: 2016-09-21
Revised: 2018-08-28
Published Online: 2018-12-16
Published in Print: 2020-06-01

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