Multiplicative function

Definition

An arithmetic function $f$ is multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all coprime positive integers $m$ and $n$.

Remarks

A multiplicative function is determined by its values at prime powers: if $n = p_1^{e_1} \cdots p_k^{e_k}$, then $f(n) = f(p_1^{e_1}) \cdots f(p_k^{e_k})$. Examples are Euler's totient function, the divisor function and the Möbius function. A function with $f(mn) = f(m) f(n)$ for all $m$ and $n$ is completely multiplicative.

Explanations

L1 human

For a multiplicative function, you can compute $f(n)$ from the prime factorization of $n$. For example, $\varphi(12) = \varphi(4) \varphi(3) = 2 \cdot 2 = 4$, because $4$ and $3$ are coprime.