# Multiplicative functions form abelian group under Dirichlet product

## Statement

Suppose  is the set of natural numbers and  is a commutative unital ring. Let  be the set of all multiplicative functions from  to , i.e., all the functions  satisfying the following:

• ,
•  whenever  are relatively prime.

Consider the Dirichlet product, a binary operation defined on functions from  to :

.

Then,  forms an abelian group under the Dirichlet product, with multiplicative identity given by the function  that takes the value  at  and  elsewhere.

## Proof

### Closure under Dirichlet product

We first need to show that the Dirichlet product is a well-defined binary operation on multiplicative functions; in other words, that a Dirichlet product of multiplicative functions is multiplicative.

Given: Multiplicative functions .

To prove:  is multiplicative.

Proof: Suppose  and  are relatively prime natural numbers. Consider:

.

Given any divisor  of ,  can be expressed uniquely as , where  and . Conversely, given  and , . Thus, we have:

.

Next, since  and  are relatively prime and  and  are also relatively prime, and  and  are multiplicative:

.

Next, we rearrange:

.

Thus:

.

### Associativity and commutativity

These follow from the fact that the Dirichlet product is associative and commutative for all functions:

### Identity element

This follows from the fact that the function  is an identity element for the Dirichlet product for all functions: Identity element for Dirichlet product is indicator function for one.

### Inverses

Given a multiplicative function , the inverse of  can be defined inductively as follows:

.

Since , this clearly satisfies:

.

Thus, . Since the multiplication is commutative,  is a two-sided inverse for .

Next, we need to verify that  is multiplicative. Fill this in later