Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n)

Stoyan Dimitrov
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 713–716
DOI: 10.7546/nntdm.2023.29.4.713-716
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Authors and affiliations

Stoyan Dimitrov
Department of Bioinformatics and Mathematical Modelling
Institute of Biophysics and Biomedical Engineering
Bulgarian Academy of Sciences
Acad. G. Bonchev Str. Bl. 105, Sofia-1113, Bulgaria

Abstract

The inequalities

\varphi^2(n)+\psi^2(n)+\sigma^2(n) \geq 3n^2+2n+3 ,
\varphi(n)\psi(n)+\varphi(n)\sigma(n)+\sigma(n)\psi(n) \geq 3n^2+2n-1

connecting \varphi(n), \psi(n) and \sigma(n)-functions are formulated and proved.

Keywords

  • Arithmetic functions φ(n), ψ(n) and σ(n)
  • Inequalities

2020 Mathematics Subject Classification

  • 11A25

References

  1. Atanassov, K. (2013). Note on φ, ψ and σ-functions. Part 6. Notes on Number Theory and Discrete Mathematics, 19(1), 22–24.
  2. Sándor, J. (2014). On certain inequalities for φ, ψ, σ  and related functions. Notes on Number Theory and Discrete Mathematics, 20(2), 52–60.
  3. Sándor, J., Mitrinović, D. S., & Crstici, B. (2006). Handbook of Number Theory I. Springer.
  4. Sándor, J., & Tóth, L. (1990). On certain number-theoretic inequalities. The Fibonacci Quarterly, 28(3), 255–258.

Manuscript history

  • Received: 10 May 2023
  • Revised: 20 July 2023
  • Accepted: 26 October 2023
  • Online First: 17 November 2023

Copyright information

Ⓒ 2023 by the Author.
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

Dimitrov, S. (2023). Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n). Notes on Number Theory and Discrete Mathematics, 29(4), 713-716, DOI: 10.7546/nntdm.2023.29.4.713-716.

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