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A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences

Abstract : Using a method due to G. J. Rieger, we show that for $1 < c < 2 $ one has, as $x$ tends to infinity $$\textrm{Card}{n \leq x : \lfloor{n^c}\rfloor} \ \textrm{ is cube-free} } = \frac{x}{\zeta(3)} + O (x^{ (c+1)/3} \log x)$$ , thus improving on a recent result by Zhang Min and Li Jinjiang.
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Submitted on : Saturday, January 19, 2019 - 8:50:15 AM
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Jean-Marc Deshouillers. A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2019, Atelier Digit_Hum, pp.127 - 132. ⟨10.46298/hrj.2019.5114⟩. ⟨hal-01986712⟩

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