Article
| Title: | On the $r$-free values of the polynomial $x^2 + y^2 + z^2 +k$ (English) |
| Author: | Chen, Gongrui |
| Author: | Wang, Wenxiao |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 955-969 |
| Summary lang: | English |
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| Category: | math |
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| 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). (English) |
| Keyword: | square-free |
| Keyword: | Salié sum |
| Keyword: | asymptotic formula |
| MSC: | 11L05 |
| MSC: | 11L40 |
| MSC: | 11N25 |
| idZBL: | Zbl 07729548 |
| idMR: | MR4632868 |
| DOI: | 10.21136/CMJ.2023.0394-22 |
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| Date available: | 2023-08-11T14:30:44Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151785 |
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