Divisor Functions

Counting and Summing Divisors

Divisor functions measure the positive divisors of an integer. They are among the first examples of arithmetic functions, because their values depend directly on the prime factorization of the input.

For a positive integer $n$ , the most basic divisor functions are

\tau(n)=\#\{d\in\mathbb{N}:d\mid n\}

and

\sigma(n)=\sum_{d\mid n} d.

The function $\tau(n)$ counts the positive divisors of $n$ . The function $\sigma(n)$ sums them.

For example, the positive divisors of $12$ are

1,2,3,4,6,12.

Hence

\tau(12)=6

and

\sigma(12)=1+2+3+4+6+12=28.

Divisors from Prime Factorization

Let

n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}

be the canonical prime factorization of $n$ .

Every positive divisor $d\mid n$ has the form

d=p_1^{\beta_1}p_2^{\beta_2}\cdots p_r^{\beta_r},

where

0\le \beta_i\le \alpha_i

for each $i$ .

Thus choosing a divisor is the same as choosing an exponent $\beta_i$ for each prime $p_i$ . For the prime $p_i$ , there are

\alpha_i+1

possible choices:

0,1,2,\ldots,\alpha_i.

Therefore

\tau(n)=(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_r+1).

For example,

72=2^3\cdot3^2.

Hence

\tau(72)=(3+1)(2+1)=12.

Sum of Divisors

The same exponent description gives a formula for $\sigma(n)$ .

Each divisor of

n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}

is obtained by choosing one power of each prime. Therefore the sum of all divisors factors as

\sigma(n) = (1+p_1+p_1^2+\cdots+p_1^{\alpha_1}) \cdots (1+p_r+p_r^2+\cdots+p_r^{\alpha_r}).

Using the finite geometric sum formula,

1+p+\cdots+p^\alpha = \frac{p^{\alpha+1}-1}{p-1},

we obtain

\sigma(n) = \prod_{i=1}^{r} \frac{p_i^{\alpha_i+1}-1}{p_i-1}.

For example,

72=2^3\cdot3^2.

Thus

\sigma(72) = (1+2+4+8)(1+3+9) = 15\cdot13 = 195.

General Divisor Sums

The divisor sum function has a useful generalization. For an integer $k\ge0$ , define

\sigma_k(n)=\sum_{d\mid n} d^k.

Then

\sigma_0(n)=\tau(n)

because every divisor contributes $1$ , and

\sigma_1(n)=\sigma(n).

n=p_1^{\alpha_1}\cdots p_r^{\alpha_r},

then

\sigma_k(n) = \prod_{i=1}^{r} (1+p_i^k+p_i^{2k}+\cdots+p_i^{\alpha_i k}).

For $k>0$ , this becomes

\sigma_k(n) = \prod_{i=1}^{r} \frac{p_i^{k(\alpha_i+1)}-1}{p_i^k-1}.

These functions occur frequently in analytic number theory and the theory of modular forms.

Multiplicativity

The functions $\tau$ , $\sigma$ , and more generally $\sigma_k$ , are multiplicative.

This means that if

\gcd(a,b)=1,

then

\tau(ab)=\tau(a)\tau(b)

and

\sigma(ab)=\sigma(a)\sigma(b).

The reason is that when $a$ and $b$ are coprime, every divisor of $ab$ can be written uniquely as a product

d=d_1d_2,

where

d_1\mid a

and

d_2\mid b.

Thus the divisor structure of $ab$ splits into the independent divisor structures of $a$ and $b$ .

Example of Multiplicativity

Let

a=8, \qquad b=9.

Since

\gcd(8,9)=1,

we have

\tau(72)=\tau(8)\tau(9).

Now

8=2^3, \qquad 9=3^2.

\tau(8)=4, \qquad \tau(9)=3.

Hence

\tau(72)=4\cdot3=12.

Similarly,

\sigma(8)=1+2+4+8=15,

and

\sigma(9)=1+3+9=13.

Therefore

\sigma(72)=15\cdot13=195.

Proper Divisors and Perfect Numbers

A proper divisor of $n$ is a positive divisor of $n$ other than $n$ itself.

The sum of proper divisors is

\sigma(n)-n.

A positive integer $n$ is called perfect if

\sigma(n)-n=n.

Equivalently,

\sigma(n)=2n.

For example,

6

is perfect because its proper divisors are

1,2,3

and

1+2+3=6.

Also,

28

is perfect because

1+2+4+7+14=28.

Perfect numbers form one of the oldest topics in number theory.

Growth of Divisor Functions

Divisor functions are irregular. Their values depend strongly on the prime factorization of $n$ .

A prime number $p$ has only two positive divisors:

1 \quad\text{and}\quad p,

\tau(p)=2.

A highly composite integer can have many divisors. For example,

360=2^3\cdot3^2\cdot5

has

\tau(360)=4\cdot3\cdot2=24

positive divisors.

Thus $\tau(n)$ is small on primes but can be large on integers with many small prime factors.

Role in Number Theory

Divisor functions translate prime factorization into numerical invariants. They measure how many ways an integer can be divided and how large its divisors are collectively.

They appear in elementary number theory, average-order estimates, Dirichlet series, modular forms, and arithmetic geometry.

The main lesson is that the divisor structure of an integer is completely controlled by its prime exponents. Divisor functions make this control explicit through concrete formulas.