multiplicative function

multiplicative function

In

number theory

, a

multiplicative function

is an

arithmetic function

f

:

ℕ

→

ℂ

such that

f

⁢

(

1

)

=

1

and, for all

a

,

b

∈

ℕ

with

gcd

⁡

(

a

,

b

)

=

1

, we have

f

⁢

(

a

⁢

b

)

=

f

⁢

(

a

)

⁢

f

⁢

(

b

)

.

An arithmetic function

f

⁢

(

n

)

is said to be

completely multiplicative

if

f

⁢

(

1

)

=

1

and

f

⁢

(

a

⁢

b

)

=

f

⁢

(

a

)

⁢

f

⁢

(

b

)

holds for

all

positive integers

a

and

b

, when they are not relatively prime. In this case, the

function

is a

homomorphism

of monoids and, because of the

fundamental theorem of arithmetic

, is completely determined by its

restriction

(

http://planetmath.org/Restriction

) to

prime numbers

. Every

completely multiplicative function

is multiplicative.

Outside of number theory, the multiplicative is usually used for all functions with the property

f

⁢

(

a

⁢

b

)

=

f

⁢

(

a

)

⁢

f

⁢

(

b

)

for all

arguments

a

and

b

. This entry discusses number theoretic multiplicative functions.

Examples

Examples of multiplicative functions include many important functions in number theory, such as:

•

φ

⁢

(

n

)

: the

Euler totient function

(also denoted

ϕ

⁢

(

n

)

), counting the

totatives

of

n

;

•

μ

⁢

(

n

)

: the Möbius function, which determines the parity of the

prime factors

of

n

if

n

is

squarefree

;

•

τ

⁢

(

n

)

: the

divisor function

(also denoted

d

⁢

(

n

)

), counting the

positive

divisors

of

n

;

•

σ

⁢

(

n

)

: the

sum of divisors function

(also denoted

σ

1

⁢

(

n

)

),

summing

the positive divisors of

n

;

•

σ

k

⁢

(

n

)

: the sum of the

k

-th powers of all the positive divisors of

n

for any

complex number

k

(typically a

natural number

);

•

id

⁢

(

n

)

: the

identity function

, defined by

id

⁢

(

n

)

=

n

;

•

id

k

⁢

(

n

)

: the

power functions

, defined by

id

k

⁢

(

n

)

=

n

k

for any complex number

k

(typically a natural number);

•

1

⁢

(

n

)

: the

constant function

, defined by

1

⁢

(

n

)

=

1

;

•

ε

⁢

(

n

)

: the

convolution identity function

, defined by:

ε

⁢

(

n

)

=

∑

d

|

n

μ

⁢

(

d

)

=

{

1

if

⁢

n

=

1

0

if

⁢

n

≠

1

where

d

runs through the positive divisors of

n

.

Properties

A multiplicative function is completely determined by its values at the powers of prime numbers, a

consequence

of the fundamental theorem of arithmetic. Thus, if

n

is a

product

of powers of distinct prime numbers, say

n

=

p

a

⁢

q

b

⁢

⋯

, then

f

⁢

(

n

)

=

f

⁢

(

p

a

)

⁢

f

⁢

(

q

b

)

⁢

⋯

. This property of multiplicative functions significantly reduces the need for computation, as in the following examples for

n

=

144

=

2

4

⋅

3

2

:

σ

⁢

(

144

)

=

σ

1

⁢

(

144

)

=

σ

1

⁢

(

2

4

)

⁢

σ

1

⁢

(

3

2

)

=

(

1

1

+

2

1

+

4

1

+

8

1

+

16

1

)

⁢

(

1

1

+

3

1

+

9

1

)

=

31

⋅

13

=

403

σ

2

⁢

(

144

)

=

σ

2

⁢

(

2

4

)

⁢

σ

2

⁢

(

3

2

)

=

(

1

2

+

2

2

+

4

2

+

8

2

+

16

2

)

⁢

(

1

2

+

3

2

+

9

2

)

=

341

⋅

91

=

31031

σ

3

⁢

(

144

)

=

σ

3

⁢

(

2

4

)

⁢

σ

3

⁢

(

3

2

)

=

(

1

3

+

2

3

+

4

3

+

8

3

+

16

3

)

⁢

(

1

3

+

3

3

+

9

3

)

=

4681

⋅

757

=

3543517

Similarly, we have:

τ

⁢

(

144

)

=

τ

⁢

(

2

4

)

⁢

τ

⁢

(

3

2

)

=

(

4

+

1

)

⁢

(

2

+

1

)

=

5

⋅

3

=

15

φ

⁢

(

144

)

=

φ

⁢

(

2

4

)

⁢

φ

⁢

(

3

2

)

=

2

3

⁢

(

2

-

1

)

⁢

3

1

⁢

(

3

-

1

)

=

8

⋅

1

⋅

3

⋅

2

=

48

Convolution

Recall that, if

f

and

g

are two arithmetic functions, one defines a new arithmetic function

f

*

g

, the Dirichlet convolution (or simply convolution) of

f

and

g

, by

(

f

*

g

)

⁢

(

n

)

=

∑

d

|

n

f

⁢

(

d

)

⁢

g

⁢

(

n

d

)

,

where the sum extends over all positive divisors

d

of

n

. Some general properties of this

operation

with respect to multiplicative functions include (here the argument

n

is omitted in all functions):

•

If both

f

and

g

are multiplicative, then so is

f

*

g

(proven here (

http://planetmath.org/ElementaryResultsAboutMultiplicativeFunctionsAndConvolution

));

•

f

*

g

=

g

*

f

(proven here (

http://planetmath.org/ArithmeticFunctionsFormARing

));

•

(

f

*

g

)

*

h

=

f

*

(

g

*

h

)

(proven here (

http://planetmath.org/ArithmeticFunctionsFormARing

));

•

f

*

ε

=

ε

*

f

=

f

(proven here (

http://planetmath.org/ArithmeticFunctionsFormARing

));

•

If

f

is multiplicative, there exists a multiplicative function

g

such that

f

*

g

=

ε

(proven here (

http://planetmath.org/ElementaryResultsAboutMultiplicativeFunctionsAndConvolution

)). In other words, every multiplicative function has a

convolution inverse

that is also multiplicative.

This shows that, with respect to convolution, the multiplicative functions form an

abelian group

with

identity element

ε

. among the multiplicative functions discussed above include:

•

μ

*

1

=

ε

(the Möbius

inversion

formula

)

•

1

*

1

=

τ

•

id

*

1

=

σ

•

id

k

*

1

=

σ

k

•

ϕ

*

1

=

id

Given a completely multiplicative function

f

, its convolution inverse is

f

⁢

μ

. See this entry (

http://planetmath.org/FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction

) for a proof.

Title

multiplicative function

Canonical name

MultiplicativeFunction

Date of creation

2013-03-22 12:47:00

Last modified on

2013-03-22 12:47:00

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

50

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11A25

Related topic

EulerProduct

Related topic

ConvolutionInversesForArithmeticFunctions

Related topic

PropertyOfCompletelyMultiplicativeFunctions

Related topic

DivisorSum

Related topic

AdditiveFunction

Related topic

ProofThatEulerPhiIsAMultiplicativeFunction

Related topic

DivisorSumOfAnArithmeticFunction

Defines

multiplicative

Defines

completely multiplicative

Defines

completely multiplicative function

Defines

convolution identity function

Defines

convolution inverse

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