# Multiplicative function

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

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

*f*(*ab*) =*f*(*a*)*f*(*b*).

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)

- the indicator function of the set . This is multiplicative if the set
*C*has the property that if*a*and*b*are in*C*, gcd(*a*,*b*)=1, then*ab*is also in C. This is the case if*C*is the set of squares, cubes, or higher powers, or if*C*is the set of square-free numbers.

- 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

- (
*n*): the function defined by (*n*) = 1 if*n*= 1 and 0 otherwise, sometimes called*multiplication unit for Dirichlet convolution*or simply the*unit function*; the Kronecker delta δ_{in}; sometimes written as*u*(*n*), not to be confused with (*n*) (completely multiplicative).

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*

- (
*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

_{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_{0}(*n*) =*d*(*n*) the number of positive divisors of*n*,_{1}(*n*) = (*n*), the sum of all the positive divisors of*n*.

- (
*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).

- (
*n*), defined by (*n*) = (−1)^{$\omega$(n)}, where the additive function (*n*) is the number of distinct primes dividing*n*.

- 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) =
_{0}(144) =_{0}(2^{4})_{0}(3^{2}) = (1^{0}+ 2^{0}+ 4^{0}+ 8^{0}+ 16^{0})(1^{0}+ 3^{0}+ 9^{0}) = 5 · 3 = 15, - (144) =
_{1}(144) =_{1}(2^{4})_{1}(3^{2}) = (1^{1}+ 2^{1}+ 4^{1}+ 8^{1}+ 16^{1})(1^{1}+ 3^{1}+ 9^{1}) = 31 · 13 = 403, ^{*}(144) =^{*}(2^{4})^{*}(3^{2}) = (1^{1}+ 16^{1})(1^{1}+ 9^{1}) = 17 · 10 = 170.

Similarly, we have:

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 . Convolution is commutative, associative, and distributive over addition.

Relations among the multiplicative functions discussed above include:

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

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

### Dirichlet series for some multiplicative functions

More examples are shown in the article on Dirichlet series.

## Multiplicative function over F_{q}[X]

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

Let *h* be a polynomial arithmetic function (i.e. a function on set of monic polynomials over *A*). Its corresponding Dirichlet series define 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 *ƒ* and *g* are two polynomial arithmetic functions, one defines *ƒ* * *g*, the *Dirichlet convolution* of *ƒ* and *g*, by

where the sum extends 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 Template:Apostol IANT