For a set of primes P, let ψ (x;P) be the number of positive integers n ≤ x 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 (Equation presented).
|Number of pages||24|
|Journal||Journal fur die Reine und Angewandte Mathematik|
|State||Published - Jun 1 2020|
Bibliographical noteFunding Information:
Kaisa Matom ki was supported by Academy of Finland grants no. 137883 and 138522. Xuancheng Shao was supported by a Glasstone Research Fellowship.
© 2020 2020 Walter de Gruyter GmbH, Berlin/Boston Suomen Akatemia 137883 138522 Kaisa Matomäki was supported by Academy of Finland grants no. 137883 and 138522. Xuancheng Shao was supported by a Glasstone Research Fellowship.
ASJC Scopus subject areas
- Mathematics (all)
- Applied Mathematics