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
Full paper (PDF, 150 Kb)

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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^ν_p(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_p 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 additive function
• Completely multiplicative function
• Leibniz rule

2010 Mathematics Subject Classification

• 11A25
• 11A41

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