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Keywords:
square-free; Salié sum; asymptotic formula
square-free; Salié sum; asymptotic formula
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, Y. Ding (2022).
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