# 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:

.

### 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.