# Dirichlet product

## Contents

## Definition

Suppose is the set of natural numbers and is a commutative unital ring. Suppose are two functions. The **Dirichlet product** or **Dirichlet convolution** of and , denoted , is defined as:

.

The sum is over all *positive integers* dividing . Equivalently, it can be written as:

.

Here, the summation is restricted to the cases where both and are positive integers.

## Facts

### Ignoring the ring

Most of the functions we deal with are integer-valued. Note that there is a natural map from the integers to any commutative unital ring, and thus, any integer-valued function can be viewed as a function to for any commutative unital ring. This makes most sense when the ring has characteristic zero, so that the map from integers to it is injective.

### Abelian monoid structure

The set of all functions from to forms a monoid with respect to the Dirichlet product:

- Dirichlet product is associative: We can see that , and both are equal to:

.

- Dirichlet product is commutative: The fact that is direct from the definition, and is based on the observation that the role of the divisors and can be switched.
- Identity element for Dirichlet product is indicator function for one: The identity element for the Dirichlet product is the function , defined as , and for .

### Abelian group structure

If (where the on the left is the natural number, and the on the right is the identity element of the ring), then has a multiplicative inverse with respect to the Dirichlet product. Moreover, this inverse *also* sends to . The functions that send to , in fact, form a group under the Dirichlet product. (More generally, we can look at all functions that send to a unit).

An important subgroup of this group is the group of all multiplicative functions. A Dirichlet product of multiplicative functions is multiplicative, and the inverse of a multiplicative function is multiplicative. `Further information: Multiplicative functions form a group under Dirichlet product`

### Important functions

A complete list of commonly studied arithmetic functions is at:

Some particular important ones are:

- The identity element for Dirichlet product: Denoted , this is the
*indicator function*for : it is at and elsewhere. - The all ones function: This function sends everything to . This is denoted by . Note that although this is the identity for
*pointwise*multiplication, it is*not*the identity for the Dirichlet product. - The Mobius function: Denoted , this is the inverse of the all ones function with respect to the Dirichlet product.
- The identity function: This function sends every natural number to itself, now viewed as a ring element. This is denoted . Although this would be the identity for
*composition*, it is*not*the identity for the Dirichlet product.