The prime number theorem over integers of power-free polynomial values

Abstract

Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge 2$ 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.

10 pages. Section 5 was added to prove our main results under the assumption that the ABC conjecture is true

References in corpus (2)

• Dynamics on the number of prime divisors for additive arithmetic semigroups
• Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem