Multiplicative function

Function equal to the product of its values on coprime factors

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.

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