Dirichlet convolution

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 (invertible elements) of this ring are the arithmetic functions f with .
Specifically, Dirichlet convolution is associative,
distributive over addition
,
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 (since ), 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. (Some authors denote this as because the associated Dirichlet series is the Riemann zeta function.)
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 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 = {1, 4, 9, ...} 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 (a standard trick for these sums).
Dirichlet inverse
Examples
Given an arithmetic function its Dirichlet inverse may be calculated recursively: the value of is in terms of for .
For :
, so
. This implies that does not have a Dirichlet inverse if .
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
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 (if there are any). 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 (note that simple convergence of both series of the left hand side does not imply convergence of the right hand side!). 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 (existence of a Möbius inversion, persistence of multiplicativity, definitions of totients, Euler-type product formulas over associated primes, etc.).
Dirichlet convolution is the convolution of the incidence algebra for the positive integers ordered by divisibility.
See also
Arithmetic function
Divisor sum identities
Möbius inversion formula
References
External links
Arithmetic functions
Bilinear maps
de:Zahlentheoretische Funktion#Faltung