## Abstract

We determine the order of magnitude of H(x, y, z), the number of integers n ≤ x having a divisor in (y; z], for all x; y and z. We also study H_{r}(x, y, z), the number of integers n ≤ x having exactly r divisors in (y, z]. When r = 1 we establish the order of magnitude of H_{1}(x, y, z) for all x; y; z satisfying z ≤ x^{1/2-ε}. For every r ≥ 2, C > 1 and " ε > 0, we determine the order of magnitude of H_{r}(x, y, z) uniformly for y large and y + y/(log y)log ^{4-1-ε} ≤ z ≤ min(y^{C}, x^{1/2-ε}). As a consequence of these bounds, we settle a 1960 con-jecture of Erdos and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.

Original language | English (US) |
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Pages (from-to) | 367-433 |

Number of pages | 67 |

Journal | Annals of Mathematics |

Volume | 168 |

Issue number | 2 |

DOIs | |

State | Published - 2008 |

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty