---
tag: 04WC36XKV6
kind: statement
role: theorem
name: Multiplicativity of the divisor function
msc: [11A25]
uses: [0YEW47VJ0J, 0AP80TT660]
sources:
  - url: https://en.wikipedia.org/wiki/Divisor_function
    locator: Properties
---

## Statement

Let $k$ be a natural number.
The [[divisor function|0YEW47VJ0J]] $\sigma_k$ is a [[multiplicative function|0AP80TT660]].
That is, $\sigma_k(mn) = \sigma_k(m) \sigma_k(n)$ for all coprime positive integers $m$ and $n$.

## Remarks

The proof: for coprime $m$ and $n$, each divisor of $mn$ is a unique product $d e$ with $d \mid m$ and $e \mid n$.
The function $\sigma_k$ is not completely multiplicative: $\sigma_0(4) = 3$, but $\sigma_0(2)^2 = 4$.
At a prime power, $\sigma_k(p^e) = 1 + p^k + p^{2k} + \dots + p^{ek}$. Multiplicativity then gives $\sigma_k(n)$ for all $n$.

## Explanations

### L1 human

For example, $\sigma(4) = 7$ and $\sigma(3) = 4$, so $\sigma(12) = 28$.
This works because $4$ and $3$ share no prime factor. Each divisor of $12$ is a divisor of $4$ times a divisor of $3$.