Dirichlet product of derivative arithmetic with an arithmetic function multiplicative

TL;DR

This paper introduces a derivative concept for integers based on prime mapping and Leibniz rule, then computes its Dirichlet product with multiplicative arithmetic functions, expanding understanding of their interactions.

Contribution

It defines a novel derivative for integers and calculates its Dirichlet product with multiplicative functions, providing new insights into their algebraic properties.

Findings

Explicit formula for the Dirichlet product involving the derivative map.

Characterization of the derivative's behavior with multiplicative functions.

Potential applications in number theory and arithmetic function analysis.

Abstract

We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibniz rule. The aim of this article is to calculate the Dirichlet product of this map with a function arithmetic multiplicative.