Dirichlet convolution
In mathematics, the 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 from the positive integers to the complex numbers, the Dirichlet convolution is a new arithmetic function defined by:where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs of positive integers whose product is n.
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 ', under pointwise addition, where is defined by, and Dirichlet convolution. The multiplicative identity is the unit function ε defined by if and if. The units of this ring are the arithmetic functions f with.Specifically, Dirichlet convolution is associative,
distributes over addition
is commutative,
and has an identity element,
Furthermore, for each having, there exists an arithmetic function with, called the ' 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.
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 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 σk
- , the sum-of-divisors function
- , the number-of-divisors function
- , by Möbius inversion of the formulas for σk, σ, and d
- , proved under Euler's totient function
- , by Möbius inversion
- , from convolving 1 on both sides of
- where λ is Liouville's 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
- is an indicator function where the set is the collection of positive primes and integral powers of two.
- where is the characteristic function of the primes.
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
Dirichlet series
If f is an arithmetic function, one defines its Dirichlet series generating function byfor 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 the convolution of the incidence algebra for the positive integers ordered by divisibility.