On the square-free values of the polynomial \(x^2 + y^2 + z^2 + k\)
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.
scientific article; zbMATH DE number 7493027
scientific article