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The Lie bracket and the arithmetic derivative. (English) Zbl 1444.11012
Let $$\emptyset\ne A\subset\mathbb{P}$$. Consider a function $$D_A: \mathbb{Z}_+\to\mathbb{N}$$ satisfying $$D_A(1)=0$$. The authors prove (Lemma 6) that the following conditions are equivalent:
(i) $$[D_A,l_p]=I$$ for $$p\in A$$, and $$[D_A,l_p]=0$$ for $$p\in\mathbb{P}\setminus A$$, where $$l_p(n)=pn$$, and $$[\cdot,\cdot]$$ denotes the commutator, a.k.a. the Lie bracket;
(ii) $$D_A=\sum_{p\in A}D_p$$, where $$D_p(n)=jp^{j-1}m$$ for $$n=p^jm$$ with $$m\perp p$$ (i.e., $$\gcd{(m,p})=1$$).
The function $$D_A$$ is called the general arithmetic derivative. (It has also been called the arithmetic subderivative, see, e.g., [the reviewer et al., Ann. Math. Inform. 50, 145–157 (2019; Zbl 1444.11015)]).
The authors also study Lie bracket properties of $$D_A$$, extensions of Lemma 6, and arithmetic differential equations.

##### MSC:
 11A25 Arithmetic functions; related numbers; inversion formulas 11A41 Primes 11R27 Units and factorization
##### Keywords:
Lie bracket; arithmetic derivative
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