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.
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]
Bibliographic References
2 Documents citing this article
Sunanta Srisopha;Teerapat Srichan;Pinthira Tangsupphathawat, 2023, Divisor Problem for the Greatest Common Divisor of Integers in Piatetski-Shapiro and Beatty Sequences, Mathematica Pannonica, 29_NS3, 2, pp. 230-237, 10.1556/314.2023.00024, https://doi.org/10.1556/314.2023.00024.
Teerapat Srichan, 2022, Multiplicative functions of special type on Piatetski-Shapiro sequences, Mathematica Slovaca, 72, 5, pp. 1145-1150, 10.1515/ms-2022-0078.