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

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YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2504.00804

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 variant of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2504.00804
PDF: https://arxiv.org/pdf/2504.00804
OA Status: green
OpenAlex ID: https://openalex.org/W4417228788

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DOI: https://doi.org/10.48550/arxiv.2504.00804

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

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-04-01

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Authors: Biao Wang, Shaoyun Yi

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Landing page: https://arxiv.org/abs/2504.00804

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