The prime number theorem over integers of power-free polynomial values
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912832413499392 |
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| author | Wang, Biao Yi, Shaoyun |
| author_facet | Wang, Biao Yi, Shaoyun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The prime number theorem over integers of power-free polynomial values Wang, Biao Yi, Shaoyun Number Theory Dynamical Systems 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. |
| title | The prime number theorem over integers of power-free polynomial values |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2504.00804 |