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Peer-reviewedCC BY 4.0Mathematics

Lower Bounds on Expressions Depending on the Functions φ(n), ψ(n) and σ(n): Part 4

  • Stoyan Dimitrov1
  • Meglena Lazarova2,3
  1. 1Department of Mathematical Analysis and Differential Equations, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Blvd. St. Kliment Ohridski 8, 1000 Sofia, Bulgaria
  2. 2Department of Mathematical Modelling and Numerical Methods, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, 8 “Kliment Ohridski” Blvd., 1000 Sofia, Bulgaria
  3. 3Department of Mathematics, Informatics and Physics, Prof. Dr. Assen Zlatarov State University of Burgas, 1 Prof. Yakimov Blvd., 8000 Burgas, Bulgaria
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Abstract

In this article, we establish four lower-bound inequalities for rational expressions involving the Euler totient function φ(n), the Dedekind psi function ψ(n), and the sum-of-divisors function σ(n). The inequalities concern rational expressions involving sums of cubes, products of these arithmetic functions, and sums of pairs of these functions. The lower bounds are given explicitly as rational functions of n and are attained when n is prime. Furthermore, we formulate four conjectures that extend the established inequalities to arbitrary positive integer powers k.

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Cite

BibTeXRIS
BibTeX
@article{dimitrov2026lower,
  title = {{Lower Bounds on Expressions Depending on the Functions φ(n), ψ(n) and σ(n): Part 4}},
  author = {Stoyan Dimitrov and Meglena Lazarova},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14213935},
  url = {https://www.mdpi.com/2227-7390/14/21/3935},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Lower Bounds on Expressions Depending on the Functions φ(n), ψ(n) and σ(n): Part 4
AU  - Stoyan Dimitrov
AU  - Meglena Lazarova
PY  - 2026
DA  - 2026/10/28
JO  - Mathematics
DO  - 10.3390/math14213935
UR  - https://www.mdpi.com/2227-7390/14/21/3935
AB  - In this article, we establish four lower-bound inequalities for rational expressions involving the Euler totient function φ(n), the Dedekind psi function ψ(n), and the sum-of-divisors function σ(n). The inequalities concern rational expressions involving sums of cubes, products of these arithmetic functions, and sums of pairs of these functions. The lower bounds are given explicitly as rational functions of n and are attained when n is prime. Furthermore, we formulate four conjectures that extend the established inequalities to arbitrary positive integer powers k.
SN  - 2227-7390
ER  -