The prime number theorem over integers of power-free polynomial values
Biao Wang, Shaoyun Yi
TL;DR
This work extends Bergelson–Richter's dynamical generalization of the prime number theorem to the realm of $k$-free polynomial values $f(n)$, where $f$ is irreducible and has no fixed $k$-th power divisor. The authors develop a decomposition technique that connects sums over $k$-free values to averages along arithmetic progressions, and they prove that the dynamical limit equals the natural Euler product $\prod_p\left(1-\frac{\rho_f(p^k)}{p^k}\right)\int_X g\,d\mu$ under suitable conditions. Concrete instances are established for polynomials from the Estermann–Hooley–Heath-Brown–Browning literature, including $f(n)=n^d+c$ with explicit $k$-thresholds, and, for certain reducible polynomials, via a Beurling-type sieve. Under the ABC conjecture, the squarefree case extends to general polynomials with an explicit constant $c_f=\prod_p\left(1-\frac{\rho_f(p^2)}{p^2}\right)$ and an error $O((\log N)^{-\gamma})$, broadening the dynamical PNT paradigm to a wide class of polynomial values.
Abstract
Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is $k$-free infinitely often, if $f$ has no fixed $k$-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also holds over integers of power-free polynomial values. In this note, we establish such variants of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.