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A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences. (English) Zbl 1448.11055
M. Zhang and J. Li [Front. Math. China 12, No. 6, 1515–1525 (2017; Zbl 1418.11137)] showed that \(|\{n\leq x: \lfloor n^c \rfloor \hbox{ is cube free } \}| = \frac{x}{\zeta(3)}+ o(x^{1-\varepsilon}\)) for all \(1 < c < 11/6\), for every \(\varepsilon < 10^{-10}\). The paper under review extends the range to \(1 < c < 2\), with an improved error term \(O(x^{(c+1)/3} \log x)\).

MSC:
11B75 Other combinatorial number theory
11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
11L03 Trigonometric and exponential sums (general theory)
11L07 Estimates on exponential sums
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