Multiplicative functions form abelian group under Dirichlet product

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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