### Abstract

For any set A of natural numbers let F(x, A) denote the number of natural numbers up to x that are divisible by no element of A and let H(x, K) be the maximum of F(x, A) when A runs over the sets not containing 1 and having a sum of reciprocals not greater than K. A logarithmic asymptotic formula is given for H(x, K)-in particular it shows H(x, K) <x^{ε} for K > K_{0}(ε)-and some related problems are discussed.

Original language | English |
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Pages (from-to) | 260-268 |

Number of pages | 9 |

Journal | Journal of Number Theory |

Volume | 14 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1982 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**On the small sieve. II. sifting by composite numbers.** / Ruzsa, I.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 14, no. 2, pp. 260-268. https://doi.org/10.1016/0022-314X(82)90051-8

}

TY - JOUR

T1 - On the small sieve. II. sifting by composite numbers

AU - Ruzsa, I.

PY - 1982

Y1 - 1982

N2 - For any set A of natural numbers let F(x, A) denote the number of natural numbers up to x that are divisible by no element of A and let H(x, K) be the maximum of F(x, A) when A runs over the sets not containing 1 and having a sum of reciprocals not greater than K. A logarithmic asymptotic formula is given for H(x, K)-in particular it shows H(x, K) ε for K > K0(ε)-and some related problems are discussed.

AB - For any set A of natural numbers let F(x, A) denote the number of natural numbers up to x that are divisible by no element of A and let H(x, K) be the maximum of F(x, A) when A runs over the sets not containing 1 and having a sum of reciprocals not greater than K. A logarithmic asymptotic formula is given for H(x, K)-in particular it shows H(x, K) ε for K > K0(ε)-and some related problems are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0013546424&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013546424&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(82)90051-8

DO - 10.1016/0022-314X(82)90051-8

M3 - Article

AN - SCOPUS:0013546424

VL - 14

SP - 260

EP - 268

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -