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Arithmetic subderivatives and Leibniz-additive functions. (English) Zbl 1444.11015
Summary: We introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. In order to generalize these notions a step further, we define that an arithmetic function \(f\) is Leibniz-additive if there is a nonzero-valued and completely multiplicative function \(h_f\) satisfying \(f(mn) = f(m) h_f (n) + f(n)h_f (m)\) for all positive integers \(m\) and \(n\). We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizing the well-known bounds for the arithmetic derivative, we establish bounds for a Leibniz-additive function.

11A25 Arithmetic functions; related numbers; inversion formulas
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
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