Sagar Mandal
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 404–409
DOI: 10.7546/nntdm.2025.31.2.404-409
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Authors and affiliations
Sagar Mandal 
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur
Kalyanpur, Kanpur, Uttar Pradesh 208016, India
Abstract
In a recent paper [7], the authors introduced new arithmetic functions 
, 
related to the classical functions 
, and 
, respectively. In this note, we study the behavior of
![\[\sum_{\substack{n\leq x\\ \omega(n)=2}}(\varphi^{+}-\varphi)(n),~~~\text{and}~~\sum_{\substack{n\leq x\\ \omega(n)=2}}(\sigma^{+}-\sigma)(n),\]](https://nntdm.net/wp-content/ql-cache/quicklatex.com-e74e501cbc873c73673d1d7cad8f5ac0_l3.png "Rendered by QuickLaTeX.com")
for any real number 
.
Keywords
- Arithmetic functions
- Inequalities for arithmetic functions
- Euler’s totient function
- Sum of positive divisors function
2020 Mathematics Subject Classification
- 11A25
- 26D15
References
- Apostol, T. M. (1976). Introduction to Analytic Number Theory. Germany: Springer.
- 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.
- Dimitrov, S. (2024). Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n), II. Notes on Number Theory and Discrete Mathematics, 30(3), 547–556.
- Dimitrov, S. (2024). Inequalities involving arithmetic functions. Lithuanian Mathematical Journal, 64(4), 421–452.
- Sándor, J. (1996). On certain inequalities involving Dedekind’s arithmetical functions. Notes on Number Theory and Discrete Mathematics, 2(1), 1–4.
- Sándor, J. (2024). On certain inequalities for φ(n), ψ(n), σ(n) and related functions, II. Notes on Number Theory and Discrete Mathematics, 30(3), 575–579.
- Sándor, J., & Atanassov, K. (2024). Some new arithmetic functions. Notes on Number Theory and Discrete Mathematics, 30(4), 851–856.
Manuscript history
- Received: 24 February 2025
- Revised: 30 May 2025
- Accepted: 12 June 2025
- Online First: 16 June 2025
Copyright information
Ⓒ 2025 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).
Related papers
- 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.
- Dimitrov, S. (2024). Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n), II. Notes on Number Theory and Discrete Mathematics, 30(3), 547–556.
- Sándor, J. (1996). On certain inequalities involving Dedekind’s arithmetical functions. Notes on Number Theory and Discrete Mathematics, 2(1), 1–4.
- Sándor, J. (2024). On certain inequalities for φ(n), ψ(n), σ(n) and related functions, II. Notes on Number Theory and Discrete Mathematics, 30(3), 575–579.
- Sándor, J., & Atanassov, K. (2024). Some new arithmetic functions. Notes on Number Theory and Discrete Mathematics, 30(4), 851–856
Cite this paper
Mandal, S. (2025). A note on newly introduced arithmetic functions φ+ and σ+. Notes on Number Theory and Discrete Mathematics, 31(2), 404-409, DOI: 10.7546/nntdm.2025.31.2.404-409.
