Geodesic
On the $r$-free values of the polynomial $x^2 + y^2 + z^2 +k$
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 3, pp. 955-969.

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Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\leq x, y,z\leq H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)= c_kH^3 +O(H^{9/4+\varepsilon })$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O(H^{7/3+\varepsilon })$ obtained by G.-L. Zhou, Y. Ding (2022).
DOI : 10.21136/CMJ.2023.0394-22
Classification : 11L05, 11L40, 11N25
Keywords: square-free; Salié sum; asymptotic formula 
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Chen, Gongrui; Wang, Wenxiao. On the $r$-free values of the polynomial $x^2 + y^2 + z^2 +k$. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 3, pp. 955-969. doi : 10.21136/CMJ.2023.0394-22. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0394-22/

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