## Abstract

Let
퐹(푥) be an irreducible polynomial with integer coefficients and degree at least 2. For
푥⩾푧⩾푦⩾2, denote by
퐻퐹(푥,푦,푧) the number of integers
푛⩽푥 such that
퐹(푛) has at least one divisor *d* with
푦<푑⩽푧. We determine the order of magnitude of
퐻퐹(푥,푦,푧) uniformly for
푦+푦/log퐶푦<푧⩽푦2 and
푦⩽푥1−훿, showing that the order is the same as the order of
퐻(푥,푦,푧), the number of positive integers
푛⩽푥 with a divisor in
(푦,푧]. Here *C* is an arbitrarily large constant and
훿>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

- Mathematics(all)