Density of power-free values of polynomials

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.