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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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