On the r-free values of the polynomial x2 + y2 + z2 + k.

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 ⩽ x, y, z ⩽ H such that the polynomial x² + y² + z² + k is r-free. We obtained the asymptotic formula of R(H, r, k) for all r ⩾ 2. Our result is new even in the case r = 2. We proved that R(H, 2, k) = c_kH³ + O(H^9/4+ε), where c_k > 0 is a constant depending on k. This improves upon the error term O(H^7/3+ε) obtained by G.-L. Zhou, Y. Ding (2022). [ABSTRACT FROM AUTHOR]