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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.

11A25 Arithmetic functions; related numbers; inversion formulas
11A41 Primes
11R27 Units and factorization
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