Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence
Let $ q $ be a sufficiently large odd integer, and let $ c \in\left(1, \frac{4}{3}\right) $. We denote $ R(c; q) $ as the count of square-free numbers in the intersection of the Lehmer set and the Piatetski-Shapiro sequence. By employing additive character properties to transform congruence equation...
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AIMS Press
2024-11-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241603 |
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| author | Zhao Xiaoqing Yi Yuan |
| author_facet | Zhao Xiaoqing Yi Yuan |
| author_sort | Zhao Xiaoqing |
| collection | DOAJ |
| description | Let $ q $ be a sufficiently large odd integer, and let $ c \in\left(1, \frac{4}{3}\right) $. We denote $ R(c; q) $ as the count of square-free numbers in the intersection of the Lehmer set and the Piatetski-Shapiro sequence. By employing additive character properties to transform congruence equations and applying Kloosterman sums and methods of exponential sums, we derive a sharp asymptotic formula as $ q $ approaches infinity, which is significant for understanding the distribution properties of the Lehmer problem. |
| format | Article |
| id | doaj-art-e747d9f0a3ef4e0a8671bc08c591b121 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-e747d9f0a3ef4e0a8671bc08c591b1212025-01-23T07:53:24ZengAIMS PressAIMS Mathematics2473-69882024-11-01912335913360910.3934/math.20241603Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequenceZhao Xiaoqing0Yi Yuan1School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, ChinaSchool of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, ChinaLet $ q $ be a sufficiently large odd integer, and let $ c \in\left(1, \frac{4}{3}\right) $. We denote $ R(c; q) $ as the count of square-free numbers in the intersection of the Lehmer set and the Piatetski-Shapiro sequence. By employing additive character properties to transform congruence equations and applying Kloosterman sums and methods of exponential sums, we derive a sharp asymptotic formula as $ q $ approaches infinity, which is significant for understanding the distribution properties of the Lehmer problem.https://www.aimspress.com/article/doi/10.3934/math.20241603lehmer setpiatetski-shapiro sequencesquare-free numbersestimate methods of exponential sumasymptotic properties |
| spellingShingle | Zhao Xiaoqing Yi Yuan Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence AIMS Mathematics lehmer set piatetski-shapiro sequence square-free numbers estimate methods of exponential sum asymptotic properties |
| title | Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence |
| title_full | Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence |
| title_fullStr | Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence |
| title_full_unstemmed | Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence |
| title_short | Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence |
| title_sort | square free numbers in the intersection of lehmer set and piatetski shapiro sequence |
| topic | lehmer set piatetski-shapiro sequence square-free numbers estimate methods of exponential sum asymptotic properties |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241603 |
| work_keys_str_mv | AT zhaoxiaoqing squarefreenumbersintheintersectionoflehmersetandpiatetskishapirosequence AT yiyuan squarefreenumbersintheintersectionoflehmersetandpiatetskishapirosequence |