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

Faculty of Mathematics and Technologies of Programming

Francisk Skorina

Gomel State University

Gomel 246019

Belarus

Andrey D. Goncharenko and Irina N. Goncharenko

State Educational Establishment "Gymnasium 71"

Gomel 246036

Belarus

**Abstract:**

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*) = α*x*. As corollaries we obtain results
about the arithmetic derivative and partial arithmetic derivatives.

(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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