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Dirichlet Convolution

Arithmetic functions can be added and multiplied pointwise, but number theory has another product that is better adapted to divisibility.

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Combining Arithmetic Functions

Arithmetic functions can be added and multiplied pointwise, but number theory has another product that is better adapted to divisibility.

Let ff and gg be arithmetic functions. Their Dirichlet convolution is the arithmetic function f∗gf*g defined by

(f∗g)(n)=∑d∣nf(d)g(nd). (f*g)(n)=\sum_{d\mid n} f(d)g\left(\frac nd\right).

The sum runs over all positive divisors dd of nn.

This operation combines the values of ff and gg over the divisor structure of nn.

First Examples

Let 1\mathbf{1} denote the constant arithmetic function

1(n)=1. \mathbf{1}(n)=1.

Then

(1∗1)(n)=∑d∣n1=τ(n). (\mathbf{1}*\mathbf{1})(n) = \sum_{d\mid n}1 = \tau(n).

Thus the divisor-counting function satisfies

τ=1∗1. \tau=\mathbf{1}*\mathbf{1}.

Let id⁡(n)=n\operatorname{id}(n)=n. Then

(id⁡∗1)(n)=∑d∣nd=σ(n). (\operatorname{id}*\mathbf{1})(n) = \sum_{d\mid n} d = \sigma(n).

So

σ=id⁡∗1. \sigma=\operatorname{id}*\mathbf{1}.

These identities show that divisor functions naturally arise from convolution.

The Identity Function for Convolution

There is a special arithmetic function ε\varepsilon defined by

ε(n)={1,n=1,0,n>1. \varepsilon(n)= \begin{cases} 1, & n=1,\\ 0, & n>1. \end{cases}

It is the identity element for Dirichlet convolution.

Indeed,

(f∗ε)(n)=∑d∣nf(d)ε(nd). (f*\varepsilon)(n) = \sum_{d\mid n} f(d)\varepsilon\left(\frac nd\right).

The term ε(n/d)\varepsilon(n/d) is nonzero only when

nd=1, \frac nd=1,

that is, when

d=n. d=n.

Therefore

(f∗ε)(n)=f(n). (f*\varepsilon)(n)=f(n).

Similarly,

ε∗f=f. \varepsilon*f=f.

Commutativity and Associativity

Dirichlet convolution is commutative:

f∗g=g∗f. f*g=g*f.

This follows by replacing the divisor dd with the complementary divisor n/dn/d.

It is also associative:

(f∗g)∗h=f∗(g∗h). (f*g)*h=f*(g*h).

Associativity follows by rewriting both sides as a sum over factorizations

abc=n. abc=n.

Together with addition of arithmetic functions, Dirichlet convolution gives arithmetic functions a ring structure.

Multiplicativity

If ff and gg are multiplicative, then their Dirichlet convolution f∗gf*g is also multiplicative.

The reason is that when

gcd⁡(a,b)=1, \gcd(a,b)=1,

every divisor of abab decomposes uniquely as

d=d1d2, d=d_1d_2,

where

d1∣a d_1\mid a

and

d2∣b. d_2\mid b.

This divisor splitting allows the convolution sum over abab to factor into a product of convolution sums over aa and bb.

Thus convolution is compatible with prime factorization.

Möbius Function as an Inverse

The Möbius function satisfies

μ∗1=ε. \mu*\mathbf{1}=\varepsilon.

Indeed,

(μ∗1)(n)=∑d∣nμ(d). (\mu*\mathbf{1})(n) = \sum_{d\mid n}\mu(d).

As shown earlier,

∑d∣nμ(d)={1,n=1,0,n>1. \sum_{d\mid n}\mu(d) = \begin{cases} 1, & n=1,\\ 0, & n>1. \end{cases}

Therefore

μ∗1=ε. \mu*\mathbf{1}=\varepsilon.

So μ\mu is the Dirichlet inverse of the constant function 1\mathbf{1}.

This identity is the algebraic heart of Möbius inversion.

Euler Totient via Convolution

Euler’s totient function satisfies

φ∗1=id⁡. \varphi*\mathbf{1}=\operatorname{id}.

Equivalently,

∑d∣nφ(d)=n. \sum_{d\mid n}\varphi(d)=n.

Using the convolution inverse of 1\mathbf{1}, we can solve for φ\varphi:

φ=id⁡∗μ. \varphi=\operatorname{id}*\mu.

Thus

φ(n)=∑d∣nμ(d)nd. \varphi(n)=\sum_{d\mid n} \mu(d)\frac nd.

This formula is another way to derive

φ(n)=n∏p∣n(1−1p). \varphi(n)=n\prod_{p\mid n}\left(1-\frac1p\right).

Dirichlet Series

Dirichlet convolution corresponds to multiplication of Dirichlet series.

If

F(s)=∑n=1∞f(n)ns F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}

and

G(s)=∑n=1∞g(n)ns, G(s)=\sum_{n=1}^{\infty}\frac{g(n)}{n^s},

then, under suitable convergence conditions,

F(s)G(s)=∑n=1∞(f∗g)(n)ns. F(s)G(s) = \sum_{n=1}^{\infty}\frac{(f*g)(n)}{n^s}.

This is the divisor-sum analogue of multiplying ordinary power series.

For example,

ζ(s)2=∑n=1∞τ(n)ns. \zeta(s)^2 = \sum_{n=1}^{\infty}\frac{\tau(n)}{n^s}.

The coefficient τ(n)\tau(n) counts the number of ways to write nn as a product of two positive integers.

Invertible Arithmetic Functions

An arithmetic function ff has a Dirichlet inverse if there exists a function gg such that

f∗g=ε. f*g=\varepsilon.

Such an inverse exists exactly when

f(1)≠0. f(1)\ne0.

The inverse can be computed recursively. From

(f∗g)(1)=f(1)g(1)=1, (f*g)(1)=f(1)g(1)=1,

we get

g(1)=1f(1). g(1)=\frac1{f(1)}.

For n>1n>1, the equation

∑d∣nf(d)g(nd)=0 \sum_{d\mid n}f(d)g\left(\frac nd\right)=0

determines g(n)g(n) from earlier values.

Role in Number Theory

Dirichlet convolution is the natural multiplication law for arithmetic functions because it follows the divisor structure of integers.

It explains why divisor sums, Möbius inversion, Euler products, and Dirichlet series fit together. It also provides a compact algebraic language for many identities.

In this language, several central formulas become simple:

τ=1∗1, \tau=\mathbf{1}*\mathbf{1}, σ=id⁡∗1, \sigma=\operatorname{id}*\mathbf{1}, μ∗1=ε, \mu*\mathbf{1}=\varepsilon, φ=id⁡∗μ. \varphi=\operatorname{id}*\mu.

These identities show how arithmetic functions are built from prime factorization and divisor summation.