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

Jean-Marc Deshouillers

doi:10.46298/hrj.2019.5114

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

hrj:5114 - Hardy-Ramanujan Journal, January 23, 2019 - https://doi.org/10.46298/hrj.2019.5114

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

Authors: Jean-Marc Deshouillers ¹

1 Institut de Mathématiques de Bordeaux

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.

https://doi.org/10.46298/hrj.2019.5114

Source: HAL:hal-01986712v1

Published on: January 23, 2019

Imported 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]

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

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