---
tag: 0AP80TT660
kind: definition
name: Multiplicative function
msc: [11A25]
defines: 0NTWN57GM3
uses: [0H2EPRZC0D, 0GFKP8F8AG]
sources:
  - url: https://en.wikipedia.org/wiki/Multiplicative_function
    locator: lead section
---

## Definition

An [[arithmetic function|0H2EPRZC0D]] $f$ is multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all [[coprime|0GFKP8F8AG]] 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|0VV04M41AN]], the [[divisor function|0YEW47VJ0J]] and the [[Möbius function|0C8QENHYZW]].
A function with $f(mn) = f(m) f(n)$ for all $m$ and $n$ is [[completely multiplicative|0GS27TKF1V]].

## 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.