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
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Authors and affiliations

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


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”.


  • 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.

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