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## Squarefree density of polynomials

Acta Arithmetica MSC: Primary 11B05; Secondary 11N32 DOI: 10.4064/aa230421-27-9 Opublikowany online: 12 February 2024

#### Streszczenie

This paper is concerned with squarefree values of polynomials $\mathcal P(\mathbf x) \in \mathbb Z[x_1,\ldots ,x_s]$ where we suppose that for each $j\le s$ we have $|x_j|\le P_j$. Then we define $N_{\mathcal P} (\mathbf P) = \sum _{\substack {\mathbf x\\ |x_j|\le P_j\\ \mathcal P(\mathbf x)\not =0}} \mu (|\mathcal P(\mathbf x)|)^2$ and we are interested in its behaviour when $\min _jP_j\rightarrow \infty$, and the extent to which this can be approximated by $N_{\mathcal P} (\mathbf P) \sim 2^sP_1\ldots P_s\mathfrak S_{\mathcal P}$ where $\mathfrak S_{\mathcal P} = \prod _p \bigg ( 1-\frac {\rho _{\mathcal P}(p^2)}{p^{2s}} \bigg )\quad \text {and}\quad \rho _{\mathcal P}(d) = \mathrm{card}\,\{\mathbf x\in \mathbb Z_d^s: \mathcal P(\mathbf x) \equiv 0\ ({\rm mod}\ d)\}.$ We establish this in a number of new cases, and in particular show that if $s\ge 2$ and $\mathfrak S_{\mathcal P}=0$, then $N_{\mathcal P} (\mathbf P) =o(P_1\ldots P_s)$ as $\min _jP_j\rightarrow \infty$.

#### Autorzy

• J. M. KowalskiDepartment of Mathematics
Penn State University
University Park, PA 16802, USA
e-mail
• R. C. VaughanDepartment of Mathematics
Penn State University
University Park, PA 16802, USA
e-mail

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