The distribution of divisors of polynomials

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 $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for $y+y/\log^C y < z\le y^2$ and $y\le x^{1-\delta}$, showing that the order is the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$ with a divisor in $(y,z]$. Here $C$ is an arbitrarily large constant and $\delta>0$ is arbitrarily small.