Dirichlet product

Definition

Suppose $\mathbb {N}$ is the set of natural numbers and $R$ is a commutative unital ring. Suppose $f,g:\mathbb {N} \to R$ are two functions. The Dirichlet product or Dirichlet convolution of $f$ and $g$, denoted $(f*g)$, is defined as:

$(f*g)(n)=\sum _{d|n}f(d)g(n/d)$.

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

$(f*g)(n)=\sum _{d_{1}d_{2}=n}f(d_{1})g(d_{2})$.

Here, the summation is restricted to the cases where both $d_{1}$ and $d_{2}$ 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 $R$ 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 $\mathbb {N}$ to $R$ forms a monoid with respect to the Dirichlet product:

• Dirichlet product is associative: We can see that $f*(g*h)=(f*g)*h$, and both are equal to:

$\sum _{d_{1}d_{2}d_{3}=n}f(d_{1})g(d_{2})h(d_{3})$.

• Dirichlet product is commutative: The fact that $f*g=g*f$ is direct from the definition, and is based on the observation that the role of the divisors $d$ and $n/d$ can be switched.
• Identity element for Dirichlet product is indicator function for one: The identity element for the Dirichlet product is the function $I$, defined as $I(1)=1$, and $I(n)=0$ for $n\neq 1$.

Abelian group structure

If $f(1)=1$ (where the $1$ on the left is the natural number, and the $1$ on the right is the identity element of the ring), then $f$ has a multiplicative inverse with respect to the Dirichlet product. Moreover, this inverse also sends $1$ to $1$. The functions that send $1$ to $1$, in fact, form a group under the Dirichlet product. (More generally, we can look at all functions that send $1$ 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:

Category:Arithmetic functions

Some particular important ones are:

• The identity element for Dirichlet product: Denoted $I$, this is the indicator function for $1$: it is $1$ at $1$ and $0$ elsewhere.
• The all ones function: This function sends everything to $1$. This is denoted by $U$. Note that although this is the identity for pointwise multiplication, it is not the identity for the Dirichlet product.
• The Mobius function: Denoted $\mu$, 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 $E$. Although this would be the identity for composition, it is not the identity for the Dirichlet product.