Extensions of the class of multiplicative functions

TL;DR

This paper explores extended classes of multiplicative functions, including quasimultiplicative, semimultiplicative, and Selberg multiplicative functions, and applies these concepts to Ramanujan's sum and related functions.

Contribution

It introduces and analyzes new classes of multiplicative functions and applies them to classical sums, expanding the understanding of their properties.

Findings

Extended classes include quasimultiplicative, semimultiplicative, and Selberg multiplicative functions.

Applied these classes to Ramanujan's sum and its analogue for regular integers.

Provided new insights into the structure and properties of these sums.

Abstract

We consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicative functions as extensions of the class of multiplicative functions. We apply these concepts to Ramanujan's sum and its analogue with respect to regular integers (mod r).