Arithmetic Functions and Multiplicative Functions — Functions on the Integers

We systematically define the principal multiplicative functions (phi, sigma, mu, d), develop the ring structure of Dirichlet convolution, and give a rigorous proof of the Moebius inversion formula. We also clarify the connection to the inclusion-exclusion principle.

Folio Official

March 1, 2026

1 Definition of Arithmetic Functions

Definition 1 (Arithmetic function).

A function

f : Z_{> 0} \to C

is called an arithmetic function.

Definition 2 (Multiplicative function).

An arithmetic function

f

is multiplicative if

f (1) = 1

and

f (mn) = f (m) f (n)

whenever

g cd (m, n) = 1

. If

f (mn) = f (m) f (n)

for all positive integers

m, n

, then

f

is completely multiplicative.

2 The Principal Arithmetic Functions

Definition 3 (Divisor function and sum-of-divisors function).

For a positive integer

n

$d (n) = \sum_{d ∣ n} 1$ : the number of positive divisors of $n$ (also written $σ_{0} (n)$ ).
$σ (n) = \sum_{d ∣ n} d$ : the sum of the positive divisors of $n$ (also written $σ_{1} (n)$ ).
More generally, $σ_{k} (n) = \sum_{d ∣ n} d^{k}$ .

Theorem 4.

Both

d

and

σ

are multiplicative. For

n = p_{1}^{e_{1}} \dots p_{k}^{e_{k}}

d (n) = i = 1 \prod k (e_{i} + 1), σ (n) = i = 1 \prod k \frac{p _{i}^{e_{i} + 1} - 1}{p _{i} - 1} .

Proof.

Multiplicativity follows from the bijection between divisors of

mn

(when

g cd (m, n) = 1

) and pairs of divisors of

m

and

n

. For a prime power, the divisors of

p^{e}

are

1, p, p^{2}, \dots, p^{e}

, giving

d (p^{e}) = e + 1

and

σ (p^{e}) = 1 + p + \dots + p^{e} = (p^{e + 1} - 1) / (p - 1)

. □

Definition 5 (M\"obius function).

The Möbius function

μ : Z_{> 0} \to {- 1, 0, 1}

is defined by

μ (n) = ⎩ ⎨ ⎧ 1 (- 1)^{k} 0 if n = 1, if n = p_{1} p_{2} \dots p_{k} (a product of k distinct primes), if n has a squared prime factor.

Lemma 6.

d ∣ n \sum μ (d) = {10 if n = 1, if n > 1.

Proof.

The case

n = 1

is immediate. For

n > 1

, let

p_{1}, \dots, p_{k}

be the distinct prime factors of

n

. The divisors

d

n

with

μ (d) \neq = 0

correspond to products of subsets of

{p_{1}, \dots, p_{k}}

, so

\sum_{d ∣ n} μ (d) = \sum_{j = 0}^{k} (j k) (- 1)^{j} = (1 - 1)^{k} = 0

. □

3 Dirichlet Convolution

Definition 7 (Dirichlet convolution).

The Dirichlet convolution of arithmetic functions

f

and

g

(f * g) (n) = d ∣ n \sum f (d) g (n / d) .

Theorem 8.

Dirichlet convolution is commutative and associative, with identity element

ε (n) = [n = 1]

(the function that is

1

n = 1

and

0

elsewhere). Every arithmetic function

f

with

f (1) \neq = 0

has a Dirichlet inverse

f^{- 1}

satisfying

f * f^{- 1} = ε

Proof.

Commutativity follows from the substitution

d \leftrightarrow n / d

\sum_{d ∣ n} f (d) g (n / d) = \sum_{d ∣ n} f (n / d) g (d)

. Associativity is verified by rewriting the double sum as a triple sum

\sum_{ab c = n} f (a) g (b) h (c)

. That

f * ε = f

follows from

ε (n / d) = [d = n]

. The inverse is defined recursively:

f^{- 1} (1) = 1/ f (1)

, and for

n > 1

f^{- 1} (n) = - \frac{1}{f ( 1 )} \sum_{d ∣ n, d < n} f (n / d) f^{- 1} (d)

. □

Remark 9.

The Dirichlet convolution of two multiplicative functions is again multiplicative. Denoting by

1

the constant function

1 (n) = 1

, we have

μ * 1 = ε

; that is,

μ

is the Dirichlet inverse of

1

4 The Möbius Inversion Formula

Theorem 10 (M\"obius inversion formula).

For arithmetic functions

f

and

g

g (n) = d ∣ n \sum f (d) for all n ⟺ f (n) = d ∣ n \sum μ (d) g (n / d) for all n .

Proof.

In the language of Dirichlet convolution, the two conditions read

g = f * 1

and

f = g * μ

. Since

μ * 1 = ε

, we can pass between them using associativity: if

g = f * 1

, then

g * μ = f * 1 * μ = f * ε = f

; conversely, if

f = g * μ

, then

f * 1 = g * μ * 1 = g * ε = g

. □

Example 11.

It is known that

\sum_{d ∣ n} φ (d) = n

(classify

1, \dots, n

according to

g cd (k, n) = d

; there are

φ (n / d)

integers in each class). Applying Mö

φ (n) = \sum_{d ∣ n} μ (d) \cdot n / d = n \sum_{d ∣ n} μ (d) / d

, which is equivalent to

φ (n) = n \prod_{p ∣ n} (1 - 1/ p)

and agrees with the derivation via inclusion-exclusion.

Number Theory Algebra Textbook Arithmetic Functions Moebius Function

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Arithmetic Functions and Multiplicative Functions — Functions on the Integers

Folio Official

March 1, 2026

1 Definition of Arithmetic Functions

Definition 1 (Arithmetic function).

A function

f : Z_{> 0} \to C

is called an arithmetic function.

Definition 2 (Multiplicative function).

An arithmetic function

f

is multiplicative if

f (1) = 1

and

f (mn) = f (m) f (n)

whenever

g cd (m, n) = 1

. If

f (mn) = f (m) f (n)

for all positive integers

m, n

, then

f

is completely multiplicative.

2 The Principal Arithmetic Functions

Definition 3 (Divisor function and sum-of-divisors function).

For a positive integer

n

$d (n) = \sum_{d ∣ n} 1$ : the number of positive divisors of $n$ (also written $σ_{0} (n)$ ).
$σ (n) = \sum_{d ∣ n} d$ : the sum of the positive divisors of $n$ (also written $σ_{1} (n)$ ).
More generally, $σ_{k} (n) = \sum_{d ∣ n} d^{k}$ .

Theorem 4.

Both

d

and

σ

are multiplicative. For

n = p_{1}^{e_{1}} \dots p_{k}^{e_{k}}

d (n) = i = 1 \prod k (e_{i} + 1), σ (n) = i = 1 \prod k \frac{p _{i}^{e_{i} + 1} - 1}{p _{i} - 1} .

Proof.

Multiplicativity follows from the bijection between divisors of

mn

(when

g cd (m, n) = 1

) and pairs of divisors of

m

and

n

. For a prime power, the divisors of

p^{e}

are

1, p, p^{2}, \dots, p^{e}

, giving

d (p^{e}) = e + 1

and

σ (p^{e}) = 1 + p + \dots + p^{e} = (p^{e + 1} - 1) / (p - 1)

. □

Definition 5 (M\"obius function).

The Möbius function

μ : Z_{> 0} \to {- 1, 0, 1}

is defined by

μ (n) = ⎩ ⎨ ⎧ 1 (- 1)^{k} 0 if n = 1, if n = p_{1} p_{2} \dots p_{k} (a product of k distinct primes), if n has a squared prime factor.

Lemma 6.

d ∣ n \sum μ (d) = {10 if n = 1, if n > 1.

Proof.

The case

n = 1

is immediate. For

n > 1

, let

p_{1}, \dots, p_{k}

be the distinct prime factors of

n

. The divisors

d

n

with

μ (d) \neq = 0

correspond to products of subsets of

{p_{1}, \dots, p_{k}}

, so

\sum_{d ∣ n} μ (d) = \sum_{j = 0}^{k} (j k) (- 1)^{j} = (1 - 1)^{k} = 0

. □

3 Dirichlet Convolution

Definition 7 (Dirichlet convolution).

The Dirichlet convolution of arithmetic functions

f

and

g

(f * g) (n) = d ∣ n \sum f (d) g (n / d) .

Theorem 8.

Dirichlet convolution is commutative and associative, with identity element

ε (n) = [n = 1]

(the function that is

1

n = 1

and

0

elsewhere). Every arithmetic function

f

with

f (1) \neq = 0

has a Dirichlet inverse

f^{- 1}

satisfying

f * f^{- 1} = ε

Proof.

Commutativity follows from the substitution

d \leftrightarrow n / d

\sum_{d ∣ n} f (d) g (n / d) = \sum_{d ∣ n} f (n / d) g (d)

. Associativity is verified by rewriting the double sum as a triple sum

\sum_{ab c = n} f (a) g (b) h (c)

. That

f * ε = f

follows from

ε (n / d) = [d = n]

. The inverse is defined recursively:

f^{- 1} (1) = 1/ f (1)

, and for

n > 1

f^{- 1} (n) = - \frac{1}{f ( 1 )} \sum_{d ∣ n, d < n} f (n / d) f^{- 1} (d)

. □

Remark 9.

The Dirichlet convolution of two multiplicative functions is again multiplicative. Denoting by

1

the constant function

1 (n) = 1

, we have

μ * 1 = ε

; that is,

μ

is the Dirichlet inverse of

1

4 The Möbius Inversion Formula

Theorem 10 (M\"obius inversion formula).

For arithmetic functions

f

and

g

g (n) = d ∣ n \sum f (d) for all n ⟺ f (n) = d ∣ n \sum μ (d) g (n / d) for all n .

Proof.

In the language of Dirichlet convolution, the two conditions read

g = f * 1

and

f = g * μ

. Since

μ * 1 = ε

, we can pass between them using associativity: if

g = f * 1

, then

g * μ = f * 1 * μ = f * ε = f

; conversely, if

f = g * μ

, then

f * 1 = g * μ * 1 = g * ε = g

. □

Example 11.

It is known that

\sum_{d ∣ n} φ (d) = n

(classify

1, \dots, n

according to

g cd (k, n) = d

; there are

φ (n / d)

integers in each class). Applying Mö

φ (n) = \sum_{d ∣ n} μ (d) \cdot n / d = n \sum_{d ∣ n} μ (d) / d

, which is equivalent to

φ (n) = n \prod_{p ∣ n} (1 - 1/ p)

and agrees with the derivation via inclusion-exclusion.

Number Theory Algebra Textbook Arithmetic Functions Moebius Function

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Arithmetic Functions and Multiplicative Functions — Functions on the Integers

1 Definition of Arithmetic Functions

2 The Principal Arithmetic Functions

3 Dirichlet Convolution

4 The Möbius Inversion Formula

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Arithmetic Functions and Multiplicative Functions — Functions on the Integers

1 Definition of Arithmetic Functions

2 The Principal Arithmetic Functions

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