# Dirichlet product

## 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.