Hardy-Ramanujan Journal |

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

Accepted 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
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Mathematics [math],
[
MATH.MATH-NT
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Mathematics [math]/Number Theory [math.NT]

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