On the multiplicative group generated by \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N} \Big\}. V

I. Kátai and B. M. Phong
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 348–353
DOI: 10.7546/nntdm.2023.29.2.348-353
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Authors and affiliations

I. Kátai
Department of Computer Algebra, University of Eötvös Loránd
1117 Budapest, Hungary

B. M. Phong
Department of Computer Algebra, University of Eötvös Loránd
1117 Budapest, Hungary

Abstract

Let f,g be completely multiplicative functions, \vert f(n)\vert=\vert g(n)\vert =1 (n\in\mathbb{N}). Assume that

    \[{1\over {\log x}}\sum_{n\le x}{\vert g([\sqrt{2}n])-Cf(n)\vert\over n}\to 0 \quad (x\to\infty).\]

Then

    \[f(n)=g(n)=n^{i\tau},\quad C=(\sqrt{2})^{i\tau}, \tau\in \mathbb{R}.\]

Keywords

  • Completely multiplicative functions
  • Multiplicative group

2020 Mathematics Subject Classification

  • 11K65
  • 11N37
  • 11N64

References

  1. Kátai, I. (1984). Multiplicative functions with regularity properties II. Acta Mathematica Hungarica, 43(1–2), 105–130.
  2. Kátai, I., & Phong, B. M. (2015). On the multiplicative group generated by \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N} \Big\}. Acta Mathematica Hungarica, 145 (1), 80–87.
  3. Kátai, I., & Phong, B. M. (2015). On the multiplicative group generated by \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N} \Big\} II. Acta Scientiarum Mathematicarum (Szeged), 81(3–4), 431–436.
  4. Kátai, I., & Phong, B. M. (2015). On the multiplicative group generated by \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N} \Big\} III. Acta Mathematica Hungarica, 147, 247–254.
  5. Kátai, I., & Phong, B. M. (2014–2015). On the multiplicative group generated by
    \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N} \Big\} IV. Mathematica Pannonica, 25(1), 105–112.
  6. Klurman, O. (2017). Correlations of multiplicative functions and applications. Compositio Mathematica, 153, 1620–1657.
  7. Klurman, O., & Mangerel, A. (2018). Rigidity theorems for multiplicative functions. Mathematische Annalen, 372(1–2), 651–697.
  8. Matomäki, K., & Radziwill, M. (2016). Mulitplicative functions in short intervals. Annals of Mathematics, 183(3), 1015–1056.
  9. Tao, T. (2016). The logarithmically averaged Chowla and Elliott conjectures for two-point correlations. Forum of Mathematics, Pi, 4, e8, 36 pages.

Manuscript history

  • Received: 31 January 2023
  • Accepted: 11 May 2023
  • Online First: 12 May 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Kátai, I. & Phong, B. M. (2023). On the multiplicative group generated by \Big\{{[\sqrt {2}n]\over n}~\mid~n\in\mathbb{N}\Big\}. V. Notes on Number Theory and Discrete Mathematics, 29(2), 348-353, DOI: 10.7546/nntdm.2023.29.2.348-353.

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