An extension of Dedekind’s arithmetic function – II

Krassimir Atanassov and József Sándor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 610–621
DOI: 10.7546/nntdm.2026.32.3.610-621
Full paper (PDF, 262 Kb)

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Authors and affiliations

Krassimir Atanassov
Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Block 105, 1113 Sofia, Bulgaria

József Sándor
Department of Mathematics, Babeș-Bolyai University
Str. Kogalniceanu 1, 400084 Cluj-Napoca, Romania

Abstract

We obtain refinements and new results related to an extension of Dedekind’s arithmetic function, introduced in [1].

Keywords

• Arithmetic functions
• Schemmel’s totient
• Euler totient
• Dedekind’s function
• Inequalities for sums and products
• Asymptotic results on arithmetic functions

2020 Mathematics Subject Classification

• 11A25
• 11N37
• 26D15

References

1. Atanassov, K. (2026). An extension of Dedekind’s arithmetic function. Notes on Number Theory and Discrete Mathematics, 32(2), 439–450.
2. Dickson, L. E. (1919). History of the Theory of Numbers, I. Reprinted: AMS Chelsea Publishing, Providence, 1999.
3. Deaconescu, M. (2000). Adding units (mod n). Elemente der Mathematik, 55(3), 123–127.
4. Lehmer, D. N. (1929). On the congruences with certain magic squares. Transactions of the American Mathematical Society, 31(3), 529–531.
5. Nageswara Rao, K. (1961). On extension of Euler’s function. The Mathematics Student, 29(3–4), 121–126.
6. Sándor, J. (1996). On certain inequalities involving Dedekind’s arithmetical function. Notes on Number Theory and Discrete Mathematics, 2(1), 1–4.
7. Sándor, J. (2010). Two arithmetic inequalities. Advanced Studies in Contemporary Mathematics, 20(2), 197–202.
8. Sándor, J. (2020). On the equation φ(n) + d(n) = n and related inequalities. Notes on Number Theory and Discrete Mathematics, 26(3), 1–4.
9. Sándor, J. (2024). On certain inequalities for σ, φ, ψ and related functions, II. Notes on Number Theory and Discrete Mathematics, 30(3), 575–579.
10. Sándor, J., & Atanassov, K. (2025). Inequalities with some arithmetic functions. Mathematics, 13(8), Article ID 1253.
11. Sándor, J., & Crstici, B. (2004). Handbook of Number Theory. II. Springer Verlag, Berlin.
12. Sándor, J., & Gryszka, K. (2025). On certain inequalities for φ, ψ, σ and related functions, III. Notes on Number Theory and Discrete Mathematics, 31(2), 361–369.
13. Sándor, J., Mitrinović, D. S., & Crstici. B. (2005). Handbook of Number Theory. I. Springer Verlag, New York.
14. Sándor, J., & Tóth, L. (1989). An asymptotic formula concerning a generalized Euler function. The Fibonacci Quarterly, 27(2), 176–180.
15. Schemmel, V. (1869). Über relative Primzahlen. Journal fur die Reine und Angewandte Mathematik, 70, 191–192.
16. Stevens, H. (1971). Generalization of the Euler φ-function. Duke Mathematical Journal, 38, 181–186.

Manuscript history

• Received: 20 September 2026
• Revised: 28 October 2026
• Accepted: 31 October 2026
• Online First: 31 October 2026

Copyright information

Ⓒ 2026 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).