Multiplicative Function
Definition
An arithmetic function \(f\) is said to be multiplicative if for any \(a, b\) which are coprime integers
\[ f(ab) = f(a)f(b).\]
This definition is used a lot in number theory, with the divisor counting function, sum of divisor function and Euler's totient function all being multiplicative.
In a more general context, sometimes the word multiplicative is used to refer to what is known as a completely multiplicative function, which drops the coprimacy condition and hence allows the definition to make sense on any set which defines multiplication.