Wikipedia / Dirichlet convolution
Wikipedia / Dirichlet convolution
In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
are two arithmetic functions is a new arithmetic function defined by:
where the sum extends over all positive divisors.
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 otherwise. The units. Since the non-units are closed under addition, this ring is local.
Specifically, Dirichlet convolution is associative,
distributive over addition
,
,
and has an identity element,
.
, called the Dirichlet inverse.
The Dirichlet convolution of two multiplicative functions), 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.
. Given a completely multiplicative function. 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 not the identity. (Some authors denote this because the associated Dirichlet series is the Riemann zeta function.)
is a set indicator function, otherwise 0.
is the identity function with value n.
is the k.
The following relations hold:
, is the Möbius function (see proof). Hence:
, 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 τ.
, proved under Euler's totient function.
, by Möbius inversion.
.
where λ is Liouville's function.
is Pillai's arithmetical function, also known as the gcd-sum function.
where Sq = {1, 4, 9, ...} is the set of squares.
is the prime omega function counting distinct prime factors of n.
, the characteristic function of the prime powers.
is the characteristic function of the primes.
This last identity shows that the prime-counting function is given by the summatory function
is the Mertens function 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
.
:
, so
.
:
,
,
:
,
,
:
,
,
,
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.
.
A multiplicative function f is completely multiplicative.
If f is completely multiplicative 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 :
convoluted with itself k 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 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.
See also
Notes
Further reading
External links