On the \(r\)-free values of the polynomial \(x^2+y^2+z^2+k\). (English) Zbl 07729548

Summary: 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 and Y. Ding [J. Number Theory 236, 308–322 (2022; Zbl 1490.11096)].


11N25 Distribution of integers with specified multiplicative constraints
11L05 Gauss and Kloosterman sums; generalizations
11L40 Estimates on character sums


Zbl 1490.11096
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