Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.3

Leibniz-Additive Functions on UFD's

Viachaslau I. Murashka
Faculty of Mathematics and Technologies of Programming
Francisk Skorina
Gomel State University
Gomel 246019

Andrey D. Goncharenko and Irina N. Goncharenko
State Educational Establishment "Gymnasium 71"
Gomel 246036


Recall that an arithmetic function f is called an L-additive function with respect to a completely multiplicative function h if f(mn) = f(m)h(n) + f(n)h(m) holds for all m and n. We study L-additive functions in the fields of fractions of unique factorization domains (UFD). In particular, we describe all L-additive functions over given UFD such that these functions can be extended to its field of fractions. We find the exact formula for an L-additive function in the terms of prime elements. For a given L-additive function f(x) we study the properties of the sequence (f(k)(x))k≥1 and solutions of the equation f(x) = αx. As corollaries we obtain results about the arithmetic derivative and partial arithmetic derivatives.

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(Concerned with sequences A000040 A003415.)

Received June 12 2020; revised versions received August 22 2020; August 25 2020. Published in Journal of Integer Sequences, October 17 2020.

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