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Research Article | Open Access

Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence

Zhao Xiaoqing, Yi Yuan( )
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
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Abstract

Let q be a sufficiently large odd integer, and let c∈(1,43). 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.

Keywords

Lehmer setPiatetski-Shapiro sequencesquare-free numbersestimate methods of exponential sumasymptotic properties
CLC number: 11B83, 11L05, 11N69

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AIMS Mathematics
Volume 9 Issue 12,
December 2024
Pages 33591-33609
DOI: 10.3934/math.20241603

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Cite this article:
Xiaoqing Z, Yuan Y. Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence. AIMS Mathematics, 2024, 9(12): 33591-33609. https://doi.org/10.3934/math.20241603

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Received: 29 August 2024
Revised: 12 November 2024
Accepted: 14 November 2024
Published: 15 December 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)