Editing Multiplicative function (section)

== Examples ==

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

* <math>1(n)</math>: the constant function defined by <math>1(n)=1</math>

* <math>\operatorname{Id}(n)</math>: the [[identity function]], defined by <math>\operatorname{Id}(n)=n</math>

* <math>\operatorname{Id}_k(n)</math>: the power functions, defined by <math>\operatorname{Id}_k(n)=n^k</math> for any complex number <math>k</math>. As special cases we have 
** <math>\operatorname{Id}_0(n)=1(n)</math>, and
** <math>\operatorname{Id}_1(n)=\operatorname{Id}(n)</math>.

* <math>\varepsilon(n)</math>: the function defined by <math>\varepsilon(n)=1</math> if <math>n=1</math> and <math>0</math> otherwise; this is the [[unit function]], so called because it is the multiplicative identity for [[Dirichlet convolution]]. Sometimes written as <math>u(n)</math>; not to be confused with <math>\mu(n)</math>.

* <math>\lambda(n)</math>: the [[Liouville function]], <math>\lambda(n)=(-1)^{\Omega(n)}</math>, where <math>\Omega(n)</math> is the total number of primes (counted with multiplicity) dividing <math>n</math>

The above functions are all completely multiplicative.

* <math>1_C(n)</math>: the [[indicator function]] of the set <math>C\subseteq \Z</math>. This function is multiplicative precisely when <math>C</math> is closed under multiplication of coprime elements. There are also other sets (not closed under multiplication) that give rise to such functions, such as the set of [[square-free]] numbers.

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

* <math>\gcd(n,k)</math>: the [[greatest common divisor]] of <math>n</math> and <math>k</math>, as a function of <math>n</math>, where <math>k</math> is a fixed integer

* <math>\varphi(n)</math>: [[Euler's totient function]], which counts the positive integers [[coprime]] to (but not bigger than) <math>n</math>

* <math>\mu(n)</math>: the [[Möbius function]], the parity (<math>-1</math> for odd, <math>+1</math> for even) of the number of prime factors of [[square-free integer|square-free]] numbers; <math>0</math> if <math>n</math> is not square-free

* <math>\sigma_k(n)</math>: the [[divisor function]], which is the sum of the <math>k</math>-th powers of all the positive divisors of <math>n</math> (where <math>k</math> may be any [[complex number]]). As special cases we have
** <math>\sigma_0(n)=d(n)</math>, the number of positive [[divisor]]s of <math>n</math>,
** <math>\sigma_1(n)=\sigma(n)</math>, the sum of all the positive divisors of <math>n</math>.

*<math>\sigma^*_k(n)</math>: the sum of the <math>k</math>-th powers of all [[unitary divisor]]s of <math>n</math> 
::<math>\sigma_k^*(n) \,=\!\!\sum_{d \,\mid\, n \atop \gcd(d,\,n/d)=1} \!\!\! d^k</math>

* <math>a(n)</math>: the number of non-isomorphic [[abelian groups]] of order <math>n</math>

* <math>\gamma(n)</math>, defined by <math>\gamma(n) = (-1)^{\omega(n)}</math>, where the [[additive function]] <math>\omega(n)</math> is the number of distinct primes dividing <math>n</math>
* <math>\tau(n)</math>: the [[Ramanujan tau function]]
* All [[Dirichlet character]]s are completely multiplicative functions, for example
** <math>(n/p)</math>, the [[Legendre symbol]], considered as a function of <math>n</math> where <math>p</math> is a fixed [[prime number]]

An example of a non-multiplicative function is the arithmetic function <math>r_2(n)</math>, the number of representations of <math>n</math> as a sum of squares of two integers, [[positive number|positive]], [[negative number|negative]], or [[0 (number)|zero]], where in counting the number of ways, reversal of order is allowed. For example:

{{block indent|em=1.2|text=1 = 1<sup>2</sup> + 0<sup>2</sup> = (−1)<sup>2</sup> + 0<sup>2</sup> = 0<sup>2</sup> + 1<sup>2</sup> = 0<sup>2</sup> + (−1)<sup>2</sup>}}

and therefore <math>r_2(1)=4\neq 1</math>. This shows that the function is not multiplicative. However, <math>r_2(n)/4</math> is multiplicative.

In the [[On-Line Encyclopedia of Integer Sequences]], sequences of values of a multiplicative function have the keyword "mult".<ref>{{cite web | url=http://oeis.org/search?q=keyword:mult | title=Keyword:mult - OEIS }}</ref>

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