Divisor Function σ(n)

Statement

The divisor function (or sum-of-divisors function) $\sigma(n)$ is defined for positive integers $n$ by

More generally, $\sigma_k(n) = \sum_{d \mid n} d^k$; the case $k = 1$ is the most classical.

Key property: $\sigma$ is multiplicative — for coprime $m, n$:

A number $n$ is perfect if $\sigma(n) = 2n$ (equivalently, the sum of proper divisors equals $n$).

Visualization

Values of $\sigma(n)$ for $n = 1$ to $20$. Perfect numbers ($\sigma(n) = 2n$) are starred:

$n$	Divisors of $n$	$\sigma(n)$	Note
1	1	1
2	1, 2	3
3	1, 3	4
4	1, 2, 4	7
5	1, 5	6
6	1, 2, 3, 6	12	perfect $\star$
7	1, 7	8
8	1, 2, 4, 8	15
9	1, 3, 9	13
10	1, 2, 5, 10	18
11	1, 11	12
12	1, 2, 3, 4, 6, 12	28
13	1, 13	14
14	1, 2, 7, 14	24
15	1, 3, 5, 15	24
16	1, 2, 4, 8, 16	31
17	1, 17	18
18	1, 2, 3, 6, 9, 18	39
19	1, 19	20
20	1, 2, 4, 5, 10, 20	42
28	1, 2, 4, 7, 14, 28	56	perfect $\star$

Multiplicativity check: $\sigma(4) = 7$, $\sigma(3) = 4$, $\sigma(12) = \sigma(4 \cdot 3) = 7 \cdot 4 = 28$. $\checkmark$

Proof Sketch

1. Multiplicativity. For coprime $m, n$, every divisor of $mn$ factors uniquely as $d_1 d_2$ with $d_1 \mid m$, $d_2 \mid n$. Hence $\sum_{d \mid mn} d = \bigl(\sum_{d_1 \mid m} d_1\bigr)\bigl(\sum_{d_2 \mid n} d_2\bigr) = \sigma(m)\sigma(n)$.
2. Prime powers. $\sigma(p^k) = 1 + p + p^2 + \cdots + p^k = (p^{k+1}-1)/(p-1)$.
3. General formula. By multiplicativity, for $n = \prod p_i^{e_i}$: $\sigma(n) = \prod_i \sigma(p_i^{e_i}) = \prod_i \frac{p_i^{e_i+1}-1}{p_i-1}$.
4. Perfect numbers. Even perfect numbers have the form $2^{p-1}(2^p - 1)$ where $2^p - 1$ is a Mersenne prime (Euler's theorem).

Connections

$\sigma$ is the canonical example of a multiplicative functionMultiplicative Functions$$\gcd(m,n)=1 \Rightarrow f(mn)=f(m)f(n)$$Arithmetic functions that split over coprime inputs: f(mn) = f(m)f(n) when gcd(m,n) = 1Read more →. The Möbius InversionMöbius Inversion$$f(n) = \sum_{d \mid n} \mu(d)\, g\!\left(\frac{n}{d}\right)$$If g equals the Dirichlet convolution of f with the constant 1, then f recovers via the Möbius functionRead more → formula expresses $\sigma$ as a Dirichlet convolution $\sigma = \mathrm{id} * \mathbf{1}$, connecting it to the theory of Euler's Totient FunctionEuler's Totient Function$$a^{\varphi(n)} \equiv 1 \pmod{n}$$φ(n) counts integers up to n coprime to n, and governs modular exponentiationRead more →.

import Mathlib.NumberTheory.ArithmeticFunction.Misc

open ArithmeticFunction in
/-- σ is multiplicative: isMultiplicative_sigma from Mathlib. -/
theorem sigma_multiplicative {k : ℕ} :
    IsMultiplicative (σ k) :=
  isMultiplicative_sigma

open ArithmeticFunction in
/-- σ₁(6) = 12: sum of divisors of 6 is 1+2+3+6 = 12.
    We verify via the explicit divisor set and sigma_one_apply. -/
theorem sigma_6 : σ 1 6 = 12 := by
  rw [sigma_one_apply]
  decide

open ArithmeticFunction in
/-- 6 is perfect: σ₁(6) = 2·6. -/
theorem six_is_perfect : σ 1 6 = 2 * 6 := by
  rw [sigma_one_apply]
  decide