Density of power-free values of polynomials

TL;DR

This paper derives asymptotic formulas for counting the number of polynomial values that are free of k-th powers, extending previous results and employing a novel sieving method for prime variables.

Contribution

It generalizes existing work on power-free polynomial values by establishing new asymptotic formulas for all variables, including primes, using a different sieving approach.

Findings

Asymptotic formulas for k-free polynomial values for degree d ≥ 2

Extension to prime variables in the counting problem

Use of a new sieving technique for the middle range of primes

Abstract

We establish asymptotic formulae for the number of $k$-free values of polynmilas $F(x_1,\cdots,x_n)\in\mathbb{Z}[x_1,\cdots,x_n]$ of degree $d\geq 2$ for any $n\geq 1$, including when the variables are prime, as long as $k\geq (3d+1)/4$. Thus we generalize a work of Browning, while we use a different sieveing technique for the middle range of primes.