Dirichlet convolution

In mathematics, Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.

Definition

If are two arithmetic functions, their Dirichlet convolution is a new arithmetic function defined by:
where the sum extends over all positive divisors of, or equivalently over all distinct pairs of positive integers whose product is.
This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients:

Properties

The set of arithmetic functions forms a commutative ring, the , with addition given by pointwise addition and multiplication by Dirichlet convolution. The multiplicative identity is the unit function defined by if and otherwise. The units of this ring are the arithmetic functions with. Since the non-units are closed under addition, this ring is local.
Specifically, Dirichlet convolution is associative,
distributive over addition
commutative,
and has an identity element,
Furthermore, for each function having, there exists another arithmetic function satisfying, called the Dirichlet inverse of.
The Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring. Beware however that the sum of two multiplicative functions is not multiplicative, so the subset of multiplicative functions is not a subring of the Dirichlet ring. The article on multiplicative functions lists several convolution relations among important multiplicative functions.
Another operation on arithmetic functions is pointwise multiplication: is defined by. Given a completely multiplicative function, pointwise multiplication by distributes over Dirichlet convolution:. The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.

Properties and examples

In these formulas, we use the following arithmetical functions:

is the multiplicative identity:, otherwise 0.
is the constant function with value 1: for all. Keep in mind that is not the identity.
for is a set indicator function: iff, otherwise 0.
is the identity function with value n: .
is the kth power function: .

The following relations hold:

, the Dirichlet inverse of the constant function is the Möbius function. Hence:
if and only if, the Möbius inversion formula.
, the kth-power-of-divisors sum function σ_k.
, the sum-of-divisors function.
, the number-of-divisors function.
, by Möbius inversion of the formulas for σ_k, σ, and τ.
, proved under Euler's totient function.
, by Möbius inversion.
, from convolving 1 on both sides of.
where λ is Liouville's function.
, where is Pillai's arithmetical function, also known as the gcd-sum function.
where Sq = is the set of squares.
, Jordan's totient function.
, where is von Mangoldt's function.
where is the prime omega function counting distinct prime factors of n.
, the characteristic function of the prime powers.
where is the characteristic function of the primes.

This last identity shows that the prime-counting function is given by the summatory function
where is the Mertens function and is the distinct prime factor counting function from above. This expansion follows from the identity for the sums over Dirichlet convolutions given on the divisor sum identities page.

Dirichlet inverse

Examples

Given an arithmetic function its Dirichlet inverse may be calculated recursively: the value of is in terms of for.
For :
For :
For :
For :
and in general for,

Properties

The following properties of the Dirichlet inverse hold:

The function f has a Dirichlet inverse if and only if.
The Dirichlet inverse of a multiplicative function is again multiplicative.
The Dirichlet inverse of a Dirichlet convolution is the convolution of the inverses of each function:.
A multiplicative function f is completely multiplicative if and only if.
If f is completely multiplicative then whenever and where denotes pointwise multiplication of functions.

Other formulas

Arithmetic function	Dirichlet inverse:
Constant function with value 1	Möbius function μ

Liouville's function λ	Absolute value of Möbius function
Euler's totient function
The generalized sum-of-divisors function

An exact, non-recursive formula for the Dirichlet inverse of any arithmetic function f is given in Divisor sum identities. A more partition theoretic expression for the Dirichlet inverse of f is given by
The following formula provides a compact way of expressing the Dirichlet inverse of an invertible arithmetic function f :
where the expression stands for the arithmetic function convoluted with itself k times. Notice that, for a fixed positive integer, if then, this is because and every way of expressing n as a product of k positive integers must include a 1, so the series on the right hand side converges for every fixed positive integer ''n.''

Dirichlet series

If f is an arithmetic function, the Dirichlet series generating function is defined by
for those complex arguments s for which the series converges. The multiplication of Dirichlet series is compatible with Dirichlet convolution in the following sense:
for all s for which both series of the left hand side converge, one of them at least converging
absolutely. This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform.

Related concepts

The restriction of the divisors in the convolution to unitary, bi-unitary or infinitary divisors defines similar commutative operations which share many features with the Dirichlet convolution.
Dirichlet convolution is a special case of the convolution multiplication for the incidence algebra of a poset, in this case the poset of positive integers ordered by divisibility.
The Dirichlet hyperbola method computes the summation of a convolution in terms of its functions and their summation functions.