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

**Full paper (PDF, 150 Kb)**

## 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

*D*_{S}(*n*) = *n* ∑_{p∈S} *ν*_{p}(*n*)/*p*,

where *n* = Π_{p ∈ ℙ} *p*^{νp(n)} ∈ ℤ_{+}. In particular, *D*_{ℙ}(*n*) = *D*(*n*) is the arithmetic derivative of *n*, and *D*_{{p}}(*n*) = *D*_{p}(*n*) is the arithmetic partial derivative of *n* with respect to *p* ∈ ℙ.

For each *p* ∈ *S*, let *f _{p}* be an arithmetic function. We define generalized arithmetic subderivative of

*n*with respect to

*S*as

*D _{S}^{f}*(

*n*) =

*n*∑

_{p∈S}

*f*(

_{p}*n*)/

*p*,

where *f* stands for the collection (*f _{p}*)

_{p∈S}of arithmetic functions. In this paper, we examine for which kind of functions

*f*the generalized arithmetic subderivative is obeys the Leibniz-rule, preserves addition, “usual multiplication” and “scalar multiplication”.

_{p}### Keywords

- Arithmetic derivative
- Arithmetic partial derivative
- Arithmetic subderivative
- Arithmetic function
- Completely additive function
- Completely multiplicative function
- Leibniz rule

### 2010 Mathematics Subject Classification

- 11A25
- 11A41

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## 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.