## Abstract

Let
𝐹(𝑥) be an irreducible polynomial with integer coefficients and degree at least 2. For
𝑥⩾𝑧⩾𝑦⩾2, denote by
𝐻𝐹(𝑥,𝑦,𝑧) the number of integers
𝑛⩽𝑥 such that
𝐹(𝑛) has at least one divisor *d* with
𝑦<𝑑⩽𝑧. We determine the order of magnitude of
𝐻𝐹(𝑥,𝑦,𝑧) uniformly for
𝑦+𝑦/log𝐶𝑦<𝑧⩽𝑦2 and
𝑦⩽𝑥1−𝛿, showing that the order is the same as the order of
𝐻(𝑥,𝑦,𝑧), the number of positive integers
𝑛⩽𝑥 with a divisor in
(𝑦,𝑧]. Here *C* is an arbitrarily large constant and
𝛿>0 is arbitrarily small.

Original language | English (US) |
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Pages (from-to) | 395-415 |

Number of pages | 21 |

Journal | Mathematika |

Volume | 66 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2020 |

## Keywords

- divisors
- polynomials

## ASJC Scopus subject areas

- General Mathematics