Deshouillers , Jean-Marc - A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences

hrj:5114 - Hardy-Ramanujan Journal, January 23, 2019
A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences

Authors: Deshouillers , Jean-Marc

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.


Source : oai:HAL:hal-01986712v1
Published on: January 23, 2019
Submitted on: January 23, 2019
Keywords: Segal-Piatetski-Shapiro sequences,cube-free numbers,estimation of trigonometric sums,discrepancy 2010 Mathematics Subject Classification 11B75,11N37,11N56,11L03,11L07, [ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]


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