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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Bibliographic Details
Published inAIMS mathematics Vol. 9; no. 12; pp. 33591 - 33609
Main Authors Xiaoqing, Zhao, Yuan, Yi
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2024
Subjects
asymptotic properties
estimate methods of exponential sum
lehmer set
piatetski-shapiro sequence
square-free numbers
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Summary: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.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.20241603