In number theory, a multiplicative function is an arithmetic function f {\displaystyle f} of a positive integer n {\displaystyle n} with the property that f ( 1 ) = 1 {\displaystyle f(1)=1} and
f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)} whenever a {\displaystyle a} and b {\displaystyle b} are coprime. An arithmetic function is said to be completely multiplicative (or totally multiplicative) if f ( 1 ) = 1 {\displaystyle f(1)=1} and f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)} holds for all positive integers a {\displaystyle a} and b {\displaystyle b} , even when they are not coprime.
Examples Some multiplicative functions are defined to make formulas easier to write:
1 ( n ) {\displaystyle 1(n)} : the constant function defined by 1 ( n ) = 1 {\displaystyle 1(n)=1}
Id ( n ) {\displaystyle \operatorname {Id} (n)} : the identity function, defined by Id ( n ) = n {\displaystyle \operatorname {Id} (n)=n}
Id k ( n ) {\displaystyle \operatorname {Id} _{k}(n)} : the power functions, defined by Id k ( n ) = n k {\displaystyle \operatorname {Id} _{k}(n)=n^{k}} for any complex number k {\displaystyle k} . As special cases we have
Id 0 ( n ) = 1 ( n ) {\displaystyle \operatorname {Id} _{0}(n)=1(n)} , and
Id 1 ( n ) = Id ( n ) {\displaystyle \operatorname {Id} _{1}(n)=\operatorname {Id} (n)} .
ε ( n ) {\displaystyle \varepsilon (n)} : the function defined by ε ( n ) = 1 {\displaystyle \varepsilon (n)=1} if n = 1 {\displaystyle n=1} and 0 {\displaystyle 0} otherwise; this is the unit function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes written as u ( n ) {\displaystyle u(n)} ; not to be confused with μ ( n ) {\displaystyle \mu (n)} .
λ ( n ) {\displaystyle \lambda (n)} : the Liouville function, λ ( n ) = ( − 1 ) Ω ( n ) {\displaystyle \lambda (n)=(-1)^{\Omega (n)}} , where Ω ( n ) {\displaystyle \Omega (n)} is the total number of primes (counted with multiplicity) dividing n {\displaystyle n}
The above functions are all completely multiplicative.
1 C ( n ) {\displaystyle 1_{C}(n)} : the indicator function of the set C ⊆ Z {\displaystyle C\subseteq \mathbb {Z} } . This function is multiplicative precisely when C {\displaystyle C} 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:
gcd ( n , k ) {\displaystyle \gcd(n,k)} : the greatest common divisor of n {\displaystyle n} and k {\displaystyle k} , as a function of n {\displaystyle n} , where k {\displaystyle k} is a fixed integer
φ ( n ) {\displaystyle \varphi (n)} : Euler's totient function, which counts the positive integers coprime to (but not bigger than) n {\displaystyle n}
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