The arithmetic derivative and Leibniz-additive functions

Abstract

An arithmetic function f is Leibniz-additive if there is a completely multiplicative function h f , i.e., h f (1) = 1 and h f (mn) = h f (m)h f (n) for all positive integers m and n, satisfyingfor all positive integers m and n. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative D; namely, D is Leibniz-additive with h D (n) = n. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function f is totally determined by the values of f and h f at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions.

Arithmetic subderivatives and Leibniz-additive functions

“…As a further extension, we defined the concept of L-additive function. For simplicity, we stated (contrary to [4]) that h f must be nonzero-valued. If we allow h f to be zero, it turns out that we then just meet extra work without gaining in results.…”

“…This paper is a sequel to [4], where we defined L-additivity without requiring that h f is nonzero-valued. We begin by showing how the values of an L-additive function f are determined in Z + by the values of f and h f at primes (Section 2) and then study under which conditions an arithmetic function f can be expressed as f = gh, where g is c-additive and h is nonzerovalued and c-multiplicative (Section 3).…”

Calculus ++ Generalized Differential and Integral Calculus

“…This linear procedure is simply a linear interpolation or a regression to the linear equation for two infinitesimally close points belonging to f (x). However, similarly to (7), from (6), a ins 0 : I → R the function a 0 instantaneous, one can write:…”

“…• Arithmetic Derivative [7] Let a.b ∈ N and p a prime number, the arithmetic derivative D(a.b) is such that:…”

A generalization of arithmetic derivative to p-adic fields and number fields

“…First, for every number field K, it is well known that O K is not necessarily a UFD. It has been proved that this idea will fail for non-UFD [4]. Second, this definition of D R (x) depends on the choice of irreducible elements set P as well as the ring.…”