# 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, a German mathematician.

## Definition

If ƒ and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function ƒ * g, the Dirichlet convolution of ƒ and g, by

{\displaystyle {\begin{aligned}(f*g)(n)&=\sum _{d\,\mid \,n}f(d)g\left({\frac {n}{d}}\right)\\&=\sum _{ab\,=\,n}f(a)g(b)\end{aligned}}}

where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integers whose product is n.

## Properties

The set of arithmetic functions forms a commutative ring, the Template:Visible anchor, under pointwise addition (i.e. f + g is defined by (f + g)(n)= f(n) + g(n)) and Dirichlet convolution. The multiplicative identity is the unit function ${\displaystyle \epsilon }$ defined by ${\displaystyle \epsilon }$(n) = 1 if n = 1 and ${\displaystyle \epsilon }$(n) = 0 if n > 1. The units (i.e. invertible elements) of this ring are the arithmetic functions f with f(1) ≠ 0.

Specifically, Dirichlet convolution is[1] associative,

(f * g) * h = f * (g * h),

f * (g + h) = f * g + f * h = (g + h) * f,

is commutative,

f * g = g * f,

and has an identity element,

f * ${\displaystyle \epsilon }$ = ${\displaystyle \epsilon }$ * f = f.

Furthermore, for each f for which f(1) ≠ 0 there exists a g such that f * g = ${\displaystyle \epsilon }$, called the Template:Visible anchor of f.

The Dirichlet convolution of two multiplicative functions is again multiplicative, and every multiplicative function has a Dirichlet inverse that is also multiplicative. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

Given a completely multiplicative function f then f (g*h) = (f g)*(f h), where juxtaposition represents pointwise multiplication.[2] The convolution of two completely multiplicative functions is a fortiori multiplicative, but not necessarily completely multiplicative.

## Examples

In these formulas

${\displaystyle \epsilon }$ is the multiplicative identity. (I.e. ${\displaystyle \epsilon }$(1) = 1, all other values 0.)
1 is the constant function whose value is 1 for all n. (I.e. 1(n) = 1.) Keep in mind that 1 is not the identity.
1C, where ${\displaystyle \scriptstyle C\subset \mathbb {Z} }$ is a set is the indicator function. (I.e. 1C(n) = 1 if n ∈ C, 0 otherwise.)
Id is the identity function whose value is n. (I.e. Id(n) = n.)
Idk is the kth power function. (I.e. Idk(n) = nk.)
The other functions are defined in the article arithmetical function.
• λ * 1 = 1Sq   where Sq = {1, 4, 9, ...} is the set of squares
• d = 1 * 1   definition of the function d(n) = σ0
• 1 = d * μ
• d 3 * 1 = (d * 1)2
• (IdsJr) * Js = Js + r

## Dirichlet inverse

Given an arithmetic function ƒ its Dirichlet inverse g = ƒ−1 may be calculated recursively (i.e. the value of g(n) is in terms of g(m) for m < n) from the definition of Dirichlet inverse.

For n = 1:

(ƒ * g) (1) = ƒ(1) g(1) = ${\displaystyle \epsilon }$(1) = 1, so
g(1) = 1/ƒ(1). This implies that ƒ does not have a Dirichlet inverse if ƒ(1) = 0.

For n = 2

(ƒ * g) (2) = ƒ(1) g(2) + ƒ(2) g(1) = ${\displaystyle \epsilon }$(2) = 0,
g(2) = −1/ƒ(1) (ƒ(2) g(1)),

For n = 3

(ƒ * g) (3) = ƒ(1) g(3) + ƒ(3) g(1) = ${\displaystyle \epsilon }$(3) = 0,
g(3) = −1/ƒ(1) (ƒ(3) g(1)),

For n = 4

(ƒ * g) (4) = ƒ(1) g(4) + ƒ(2) g(2) + ƒ(4) g(1) = ${\displaystyle \epsilon }$(4) = 0,
g(4) = −1/ƒ(1) (ƒ(4) g(1) + ƒ(2) g(2)),

and in general for n > 1,

${\displaystyle g(n)={\frac {-1}{f(1)}}\sum _{\stackrel {d\,\mid \,n}{d

Since the only division is by ƒ(1) this shows that ƒ has a Dirichlet inverse if and only if ƒ(1) ≠ 0.

Here is a useful table of Dirichlet inverses of common arithmetic functions:

Arithmetic function Dirichlet inverse
Constant function equal to 1 Möbius function ${\displaystyle \mu }$
${\displaystyle n^{\alpha }}$ ${\displaystyle \mu (n)\,n^{\alpha }}$
Liouville's function ${\displaystyle \lambda }$ Absolute value of Möbius function ${\displaystyle |\mu |}$

## Dirichlet series

If f is an arithmetic function, one defines its Dirichlet series generating function by

${\displaystyle DG(f;s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}}}$

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:

${\displaystyle DG(f;s)DG(g;s)=DG(f*g;s)\,}$

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.

## Related Concepts

Template:Expand section 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.).

## References

1. Proofs of all these facts are in Chan, ch. 2
2. A proof is in the article Completely multiplicative function#Proof of pseudo-associative property.

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