Number, The Number Theory Wiki (barely begun)

From Number

Contents 1 Statement 2 Proof 2.1 Closure under Dirichlet product 2.2 Associativity and commutativity 2.3 Identity element 2.4 Inverses

Statement

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

• f(1) = 1,
• f(mn) = f(m)f(n) whenever m,n are relatively prime.

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

(f * g)(n) =	∑	f(d)g(n / d)
	d | n

.

Then, M forms an abelian group under the Dirichlet product, with multiplicative identity given by the function I that takes the value 1 at 1 and 0 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 f,g.

To prove: f * g is multiplicative.

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

(f * g)(mn) =	∑	f(d)g(mn / d)
	d | mn

.

Given any divisor d of mn, d can be expressed uniquely as d₁d₂, where d₁ | m and d₂ | n. Conversely, given d₁ | m and d₂ | n, d₁d₂ | mn. Thus, we have:

.

Next, since d₁ and d₂ are relatively prime and m / d₁ and n / d₂ are also relatively prime, and f and g are multiplicative:

.

Next, we rearrange:

.

Thus:

(f * g)(mn) = (f * g)(m)(f * g)(n).

Associativity and commutativity

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

• Dirichlet product is associative
• Dirichlet product is commutative

Identity element

This follows from the fact that the function I 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 f, the inverse of f can be defined inductively as follows:

.

Since f(1) = 1, this clearly satisfies:

.

Thus, g * f = I. Since the multiplication is commutative, g is a two-sided inverse for f.

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