Abstract
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) = ckH3 + O(H9/4+ε), where ck > 0 is a constant depending on k. This improves upon the error term O(H7/3+ε) obtained by G.-L. Zhou, Y. Ding (2022).
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This work was supported by NSFC grant 11922113.
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Chen, G., Wang, W. On the r-free values of the polynomial x2 + y2 + z2 + k. Czech Math J 73, 955–969 (2023). https://doi.org/10.21136/CMJ.2023.0394-22
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DOI: https://doi.org/10.21136/CMJ.2023.0394-22