In mathematics, the 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.
If
f,g:N\toC
(f*g)(n) = \sumd\midf(d)g\left(
| n | |
| d |
\right) = \sumab=nf(a)g(b)
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:
\left(\sumn\geq
| f(n) | |
| ns |
\right) \left(\sumn\geq
| g(n) | |
| ns |
\right) = \left(\sumn\geq
| (f*g)(n) | |
| ns |
\right).
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,[1] Dirichlet convolution is associative,
(f*g)*h=f*(g*h),
f*(g+h)=f*g+f*h
f*g=g*f
f*\varepsilon
\varepsilon*f=f
f
f(1) ≠ 0
f-1
f*f-1=\varepsilon
f
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
(f+g)(1)=f(1)+g(1)=2 ≠ 1
h
h
(f*g)h=(fh)*(gh)
In these formulas, we use the following arithmetical functions:
\varepsilon
\varepsilon(1)=1
\varepsilon(n)=\lfloor\tfrac1n\rfloor
1
1(n)=1
n
1
\zeta
1C
C\subsetN
1C(n)=1
n\inC
Id
Id(n)=n
Idk
| k | |
| Id | |
| k(n)=n |
The following relations hold:
1*\mu=\varepsilon
1
g=f*1
f=g*\mu
\sigmak=Idk*1
\sigma=Id*1
\tau=1*1
Idk=\sigmak*\mu
Id=\sigma*\mu
1=\tau*\mu
\phi*1=Id
\phi=Id*\mu
\sigma=\phi*\tau
\phi*1=Id
λ*|\mu|=\varepsilon
λ*1=1Sq
Idk*(Idk\mu)=\varepsilon
\tau3*1=(\tau*1)2
Jk*1=Idk
(IdsJr)*Js=Js
Λ*1=log
Λ
|\mu|\ast1=2\omega,
\omega(n)
\Omega\ast\mu=1l{P
\omega\ast\mu=1P
1P(n)\mapsto\{0,1\}
This last identity shows that the prime-counting function is given by the summatory function
\pi(x)=\sumn(\omega\ast\mu)(n)=
| x | |
| \sum | |
| d=1 |
\omega(d)M\left(\left\lfloor
| x | |
| d |
\right\rfloor\right)
where
M(x)
\omega
Given an arithmetic function
f
g=f-1
g(n)
g(m)
m<n
For
n=1
(f*g)(1)=f(1)g(1)=\varepsilon(1)=1
g(1)=1/f(1)
f
f(1)=0
For
n=2
(f*g)(2)=f(1)g(2)+f(2)g(1)=\varepsilon(2)=0
g(2)=-(f(2)g(1))/f(1)
For
n=3
(f*g)(3)=f(1)g(3)+f(3)g(1)=\varepsilon(3)=0
g(3)=-(f(3)g(1))/f(1)
For
n=4
(f*g)(4)=f(1)g(4)+f(2)g(2)+f(4)g(1)=\varepsilon(4)=0
g(4)=-(f(4)g(1)+f(2)g(2))/f(1)
and in general for
n>1
g(n) =
| -1 | |
| f(1) |
| \sumd\mid | f\left( | |
| d<n |
| n | |
| d |
\right)g(d).
The following properties of the Dirichlet inverse hold:[4]
(f\astg)-1=f-1\astg-1
f-1(n)=\mu(n)f(n)
(f ⋅ g)-1=f ⋅ g-1
g(1) ≠ 0
⋅
| Arithmetic function | Dirichlet inverse:[5] | ||||
|---|---|---|---|---|---|
| Constant function with value 1 | Möbius function μ | ||||
n\alpha | \mu(n)n\alpha | ||||
| Liouville's function λ | Absolute value of Möbius function | ||||
| Euler's totient function \varphi | \sumd|nd\mu(d) | ||||
| The generalized sum-of-divisors function \sigma\alpha | \sumd|nd\alpha\mu(d)\mu\left(
\right) |
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
f-1(n)=
| \Omega(n) | |
| \sum | |
| k=1 |
\left\{
| \sum | |
| {λ1+2λ2+ … +kλk=n |
\atop{λ1,λ2,\ldots,λk|n}}
| (λ1+λ2+ … +λk)! | |
| 1!2! … k! |
(-1)kf(λ1)
| 2 | |
| f(λ | |
| 2) |
…
| k\right\}. | |
| f(λ | |
| k) |
f-1
| +infty | |
| =\sum | |
| k=0 |
| (f(1)\varepsilon-f)*k | |
| f(1)k+1 |
where the expression
(f(1)\varepsilon-f)*k
f(1)\varepsilon-f
n
k>\Omega(n)
(f(1)\varepsilon-f)*k(n)=0
f(1)\varepsilon(1)-f(1)=0
If f is an arithmetic function, the Dirichlet series generating function is defined by
DG(f;s)=
| infty | |
| \sum | |
| n=1 |
| f(n) | |
| ns |
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:
DG(f;s)DG(g;s)=DG(f*g;s)
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.
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.
. Hugh L. Montgomery . Hugh Montgomery (mathematician) . Robert C. Vaughan . Robert Charles Vaughan (mathematician) . Multiplicative number theory I. Classical theory . Cambridge tracts in advanced mathematics . 97 . 2007 . 978-0-521-84903-6 . 38 . Cambridge Univ. Press . Cambridge .