# Multiplicative function

In number theory, a **multiplicative function** is an arithmetic function *f*(*n*) of a positive integer *n* with the property that *f*(1) = 1 and whenever
*a* and *b* are coprime, then

*Outside number theory, the term***multiplicative function**is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.

An arithmetic function *f*(*n*) is said to be **completely multiplicative** (or **totally multiplicative**) if *f*(1) = 1 and *f*(*ab*) = *f*(*a*)*f*(*b*) holds *for all* positive integers *a* and *b*, even when they are not coprime.

## Examples

Some multiplicative functions are defined to make formulas easier to write:

- 1(
*n*): the constant function, defined by 1(*n*) = 1 (completely multiplicative) - Id(
*n*): identity function, defined by Id(*n*) =*n*(completely multiplicative) - Id
_{k}(*n*): the power functions, defined by Id_{k}(*n*) =*n*^{k}for any complex number*k*(completely multiplicative). As special cases we have- Id
_{0}(*n*) = 1(*n*) and - Id
_{1}(*n*) = Id(*n*).

- Id
*ε*(*n*): the function defined by*ε*(*n*) = 1 if*n*= 1 and 0 otherwise, sometimes called*multiplication unit for Dirichlet convolution*or simply the*unit function*(completely multiplicative). Sometimes written as*u*(*n*), but not to be confused with*μ*(*n*) .- 1
_{C}(*n*), the indicator function of the set*C*⊂**Z**, for certain sets*C*. The indicator function 1_{C}(*n*) is multiplicative precisely when the set*C*has the following property for any coprime numbers*a*and*b*: the product*ab*is in*C*if and only if the numbers*a*and*b*are both themselves in*C*. This is the case if*C*is the set of squares, cubes, or*k*-th powers, or if*C*is the set of square-free numbers.

Other examples of multiplicative functions include many functions of importance in number theory, such as:

- gcd(
*n*,*k*): the greatest common divisor of*n*and*k*, as a function of*n*, where*k*is a fixed integer. - (
*n*): Euler's totient function , counting the positive integers coprime to (but not bigger than)*n* *μ*(*n*): the Möbius function, the parity (−1 for odd, +1 for even) of the number of prime factors of square-free numbers; 0 if*n*is not square-free*σ*_{k}(*n*): the divisor function, which is the sum of the*k*-th powers of all the positive divisors of*n*(where*k*may be any complex number). Special cases we have*σ*_{0}(*n*) =*d*(*n*) the number of positive divisors of*n*,*σ*_{1}(*n*) =*σ*(*n*), the sum of all the positive divisors of*n*.

*a*(*n*): the number of non-isomorphic abelian groups of order*n*.*λ*(*n*): the Liouville function,*λ*(*n*) = (−1)^{Ω(n)}where Ω(*n*) is the total number of primes (counted with multiplicity) dividing*n*. (completely multiplicative).*γ*(*n*), defined by*γ*(*n*) = (−1)^{ω(n)}, where the additive function*ω*(*n*) is the number of distinct primes dividing*n*.*τ*(*n*): the Ramanujan tau function.- All Dirichlet characters are completely multiplicative functions. For example
- (
*n*/*p*), the Legendre symbol, considered as a function of*n*where*p*is a fixed prime number.

- (

An example of a non-multiplicative function is the arithmetic function *r*_{2}(*n*) - the number of representations of *n* as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example:

- 1 = 1
^{2}+ 0^{2}= (−1)^{2}+ 0^{2}= 0^{2}+ 1^{2}= 0^{2}+ (−1)^{2}

and therefore *r*_{2}(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, *r*_{2}(*n*)/4 is multiplicative.

In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".

See arithmetic function for some other examples of non-multiplicative functions.

## 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 primes, 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}:

- d(144) =
*σ*_{0}(144) =*σ*_{0}(2^{4})*σ*_{0}(3^{2}) = (1^{0}+ 2^{0}+ 4^{0}+ 8^{0}+ 16^{0})(1^{0}+ 3^{0}+ 9^{0}) = 5 · 3 = 15, *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*^{*}(144) =*σ*^{*}(2^{4})*σ*^{*}(3^{2}) = (1^{1}+ 16^{1})(1^{1}+ 9^{1}) = 17 · 10 = 170.

Similarly, we have:

- (144)=(2
^{4})(3^{2}) = 8 · 6 = 48

In general, if *f*(*n*) is a multiplicative function and *a*, *b* are any two positive integers, then

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

## Convolution

If *f* and *g* are two multiplicative functions, one defines a new multiplicative function *f* * *g*, the *Dirichlet convolution* of *f* and *g*, by

where the sum extends over all positive divisors *d* of *n*.
With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is *ε*. Convolution is commutative, associative, and distributive over addition.

Relations among the multiplicative functions discussed above include:

*μ** 1 =*ε*(the Möbius inversion formula)- (
*μ*Id_{k}) * Id_{k}=*ε*(generalized Möbius inversion) - * 1 = Id
*d*= 1 * 1*σ*= Id * 1 = **d**σ*_{k}= Id_{k}* 1- Id = * 1 =
*σ***μ* - Id
_{k}=*σ*_{k}**μ*

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

The Dirichlet convolution of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime :

### Dirichlet series for some multiplicative functions

More examples are shown in the article on Dirichlet series.

## Multiplicative function over *F*_{q}[*X*]

*F*

_{q}[

*X*]

Let *A* = *F*_{q}[*X*], the polynomial ring over the finite field with *q* elements. *A* is a principal ideal domain and therefore *A* is a unique factorization domain.

A complex-valued function on *A* is called **multiplicative** if whenever *f* and *g* are relatively prime.

### Zeta function and Dirichlet series in *F*_{q}[*X*]

*F*

_{q}[

*X*]

Let *h* be a polynomial arithmetic function (i.e. a function on set of monic polynomials over *A*). Its corresponding Dirichlet series is defined to be

where for set if and otherwise.

The polynomial zeta function is then

Similar to the situation in **N**, every Dirichlet series of a multiplicative function *h* has a product representation (Euler product):

where the product runs over all monic irreducible polynomials *P*. For example, the product representation of the zeta function is as for the integers:

Unlike the classical zeta function, is a simple rational function:

In a similar way, If *f* and *g* are two polynomial arithmetic functions, one defines *f* * *g*, the *Dirichlet convolution* of *f* and *g*, by

where the sum is over all monic divisors *d* of *m*, or equivalently over all pairs (*a*, *b*) of monic polynomials whose product is *m*. The identity still holds.

## See also

## References

- See chapter 2 of Apostol, Tom M. (1976),
*Introduction to analytic number theory*, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001