The distribution of divisors of polynomials
Kevin Ford, Guoyou Qian

TL;DR
This paper investigates the distribution of divisors of polynomial values, establishing that their divisor count behavior mirrors that of integers, within specified ranges, for irreducible polynomials with integer coefficients.
Contribution
It extends divisor distribution results from integers to polynomial values, providing uniform estimates for the number of polynomial values with divisors in certain ranges.
Findings
Order of magnitude of $H_F(x, y, z)$ matches that of $H(x,y,z)$
Uniform estimates hold for specified ranges of $y,z,x$
Results apply to irreducible polynomials with degree at least 2
Abstract
Let be an irreducible polynomial with integer coefficients and degree at least 2. For , denote by the number of integers such that has at least one divisor with . We determine the order of magnitude of uniformly for and , showing that the order is the same as the order of , the number of positive integers with a divisor in . Here is an arbitrarily large constant and is arbitrarily small.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Advanced Mathematical Identities
