On the square-free values of the polynomial \(x^2 + y^2 + z^2 + k\)
type: Item
instance of: scholarly article
title: On the square-free values of the polynomial \(x^2 + y^2 + z^2 + k\)
zbMATH Open document ID: 1490.11096
author: Guang-Liang Zhou
author: Yuchen Ding
published in: Journal of Number Theory
publication date: 2022-03-18T00:00:00Z
review text: For any given integer \(k\), and for any odd positive integer \(\ell\), let \(\lambda_\ell(q)\) be the number of the solutions to the congruence equation \(x_1^2+\dots+x_\ell^2+k\equiv 0\pmod{q}\), with \(1\leq x_1,\dots,x_\ell\leq q\). In the paper under review, the authors prove the following asymptotic formula of square-free values represented by \(x_1^2+\dots+x_\ell^2+k\) for the case \(\ell=3\), \[ \sum_{1\leq x_1,\dots,x_\ell\leq H}\mu^2(x_1^2+\dots+x_\ell^2+k)=\prod_p\left(1-\frac{\lambda_\ell(p^2)}{p^{2\ell}}\right)H^\ell+O(H^{\ell-2/3+\varepsilon}). \] The authors mention that for other odd values of \(\ell\) the same argument works as well.
reviewed by: Mehdi Hassani
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 7493027
zbMATH Keywords: Salié sum
zbMATH Keywords: square-free numbers
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.07.022
OpenAlex ID: W3195865108
Wikidata QID: Q114156786
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DOI: 10.1016/J.JNT.2021.07.022
description: scientific article; zbMATH DE number 7493027