Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence - Xiaoqing, Zhao; Yuan, Yi (AIMS mathematics, 2024)
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.