# Generalized arithmetic subderivative

Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 2, Pages 1—7
DOI: 10.7546/nntdm.2019.25.2.1-7

## Details

### Authors and affiliations

Pentti Haukkanen Faculty of Information Technology and Communication Sciences
FI-33014 Tampere University, Finland

### Abstract

Let 0 ≠ S ⊆ ℙ. The arithmetic subderivative of n with respect to S is defined as

DS(n) = npS νp(n)/p,

where n = Πp ∈ ℙ pνp(n) ∈ ℤ+. In particular, D(n) = D(n) is the arithmetic derivative of n, and D{p}(n) = Dp(n) is the arithmetic partial derivative of n with respect to p ∈ ℙ.

For each pS, let fp be an arithmetic function. We define generalized arithmetic subderivative of n with respect to S as

DSf(n) = npS fp(n)/p,

where f stands for the collection (fp)pS of arithmetic functions. In this paper, we examine for which kind of functions fp the generalized arithmetic subderivative is obeys the Leibniz-rule, preserves addition, “usual multiplication” and “scalar multiplication”.

### Keywords

• Arithmetic derivative
• Arithmetic partial derivative
• Arithmetic subderivative
• Arithmetic function
• Completely multiplicative function
• Leibniz rule

• 11A25
• 11A41

### References

1. Apostol, T. M. (1976). Introduction to Analytic Number Theory, Springer-Verlag, New York.
2. Atanassov, K. T. (1987). New integer functions, related to φ and σ functions, Bull. Number Theory Related Topics 11 (1–3), 3–26.
3. Barbeau, E. J. (1961). Remarks on an arithmetic derivative, Canad. Math. Bull. 4 (2), 117– 122.
4. Haukkanen, P. (2012). Extensions of the class of multiplicative functions, East–West J. Math. 14 (2), 101–113.
5. Haukkanen, P., Merikoski, J. K., Mattila, M., & Tossavainen, T. (2017). The arithmetic Jacobian matrix and determinant, J. Integer Seq. 20, Article 17.9.2.
6. Haukkanen, P., Merikoski, J. K., & Tossavainen, T. (2016). On arithmetic partial differential equations, J. Integer Seq. 19, Article 16.8.6.
7. Haukkanen, P., Merikoski, J. K., & Tossavainen, T. (2018). The arithmetic derivative and Leibniz-additive functions, Notes on Number Theory and Discrete Mathematics 24 (3), 68–76.
8. Kovic, J. (2012). The arithmetic derivative and antiderivative, J. Integer Seq. 15, Article 12.3.8.
9. McCarthy, P. J. (1986). Introduction to Arithmetical Functions, Springer-Verlag, New York.
10. Merikoski, J. K., Haukkanen, P., & Tossavainen, T. (2019). Arithmetic subderivatives and Leibniz-additive functions, Ann. Math. Inform., accepted.
11. Sandor, J. & Crstici, B. (2004). Handbook of Number Theory II, Kluwer Academic, Dordrecht.
12. Sivaramakrishnan, R. (1989). Classical Theory of Arithmetic Functions, Monographs and Textbooks in Pure and Applied Mathematics 126, Marcel Dekker.
13. Ufnarovski, V. & Ahlander, B. (2003). How to differentiate a number, J. Integer Seq. 6, Article 03.3.4.

## Cite this paper

Haukkanen, P. (2019). Generalized arithmetic subderivative. Notes on Number Theory and Discrete Mathematics, 25(2), 1-7, doi: 10.7546/nntdm.2019.25.2.1-7.