Article
| Title: | On $k$-free numbers over Beatty sequences (English) |
| Author: | Zhang, Wei |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 839-847 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty $ and any constant $\varepsilon >0$, we can show that $$ \sum _{ 1\leq n\leq x \atop [\alpha n+\beta ]\in \mathcal {Q}_{k}} 1- \frac {x}{ \zeta (k)} \ll x^{k/(2k-1)+\varepsilon }+x^{1-1/(\tau +1)+\varepsilon }, $$ where $\mathcal {Q}_{k}$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$ $\varepsilon ,$ $k$ and $\beta .$ This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type $$ \sum _{1\leq h\leq H}\sum _{ 1\leq n\leq x \atop n\in \mathcal {Q}_{k}}e(\vartheta hn). $$ (English) |
| Keyword: | $k$-free number |
| Keyword: | exponential sum |
| Keyword: | Beatty sequence |
| MSC: | 11B83 |
| MSC: | 11L07 |
| idZBL: | Zbl 07729540 |
| idMR: | MR4632860 |
| DOI: | 10.21136/CMJ.2023.0304-22 |
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| Date available: | 2023-08-11T14:25:03Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151777 |
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