Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities.

A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

Definition

The divisor function σ_x(n) is defined as the sum of the x^th powers of the positive divisors of n, or

\sigma_{x}(n)=\sum_{d|n} d^x\,\! .

The notations d(n) and $\tau(n)$ (the tau function) are also used to denote σ₀(n), or the number of divisors of n. When x is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted.

\sigma_{0}(n) \equiv \tau(n) \equiv  d(n)

\sigma_{1}(n) \equiv \sigma(n)

Example

For example, σ₀(12) may be considered as the sum of the zeroth powers of the divisors of 12:

$\sigma_{0}(12)$ $= 1^0 + 2^0 + 3^0 + 4^0 + 6^0 + 12^0$
$= 1 + 1 + 1 + 1 + 1 + 1 = 6.$

while σ₁(12) is equal to the sum of the divisors' first powers:

$\sigma_{1}(12)$ $= 1^1 + 2^1 + 3^1 + 4^1 + 6^1 + 12^1$
$= 1 + 2 + 3 + 4 + 6 + 12 = 28.$

Properties

For a prime number p,

d(p) = 2

d(p^n) = n+1

\sigma(p) = p+1

because by definition, the factors of a prime number are 1 and itself. Clearly, 1 < d(n) < n and σ(n) > n for all n > 1.

The divisor function is multiplicative, but not completely multiplicative. The consequence of this is that, if we write

n = \prod_{i=1}^{r}p_{i}^{a_{i}}

where r is the number of distinct prime factors of n, p_i is the i^th prime factor, and a_i is the maximum power of p_i by which n is divisible, then we have

\sigma_x(n) = \prod_{i=1}^{r} \frac{p_{i}^{(a_{i}+1)x}-1}{p_{i}^x-1}

which is equivalent to the useful formula:

\sigma_x(n) = \prod_{i=1}^{r} \sum_{j=0}^{a_{i}} p_{i}^{j x} = \prod_{i=1}^{r} (1 + p_{i}^x + p_{i}^{2x} + ... + p_{i}^{a_i x})

An equation for calculating $\tau(n)$ is

\tau(n)=\prod_{i=1}^{r} (a_i+1)

For example, if n is 24, there are two prime factors (p₁ is 2; p₂ is 3); noting that 24 is the product of 2³×3¹, a₁ is 3 and a₂ is 1. Thus we can calculate $\tau(24)$ as so:

$\tau(24)$ $= \prod_{i=1}^{2} (a_i+1)$
$= (3 + 1)(1 + 1) = 4 \times 2 = 8.$

The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.

We also note $s(n) = \sigma(n) - n$ . This function is the one used to recognize perfect numbers which are the n for which $s(n) = n$ . If s(n) > n then n is an abundant number and if s(n) < n then n is a deficient number.

As an example, for two distinct primes p and q, let

n = pq.

Then

\phi(n) = (p-1)(q-1) = n + 1 - (p+q),

\sigma(n) = (p+1)(q+1) = n + 1 + (p+q).

In 1984, Roger Heath-Brown proved that

d(n) = d(n + 1)

will occur infinitely often.

Series relations

Two Dirichlet series involving the divisor function are:

\sum_{n=1}^{\infty} \frac{\sigma_{a}(n)}{n^s}=\zeta(s) \zeta(s-a)

and

\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s}=\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}

A Lambert series involving the divisor function is:

\sum_{n=1}^{\infty} q^n \sigma_a(n) = \sum_{n=1}^{\infty} \frac{n^a q^n}{1-q^n}

for arbitrary complex |q| ≤ 1 and a. This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.

Approximate growth rate

The behaviour of the sigma function is irregular. The growth rate of the sigma function can be expressed by:

\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\ \log \log n}=e^\gamma .

where $\gamma$ is Euler's constant. This result is Gronwall's theorem, published in 1913.

Interestingly, in 1984 Guy Robin proved that

\sigma(n) for

n > 5,040

is true if and only if the Riemann hypothesis is true (this is Robin's theorem). The largest known value that violates the inequality is n=5,040. If the Riemann hypothesis is true, there are no greater exceptions. If the hypothesis is false then there are an infinite number of values of n that violate the inequality.

A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that

\sigma(n) \le H_n + \ln(H_n)e^{H_n}

for every natural number n, where

H_n

is the nth harmonic number.

References

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9

Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, volume 1, 1996, MIT Press. ISBN 0-262-02405-5, see page 234 in section 8.8.

Robin, G. "Grandes Valeurs de la fonction somme des diviseurs et hypothèse de Riemann." J. Math. Pures Appl. 63, 187-213, 1984. Original publication of Robin's theorem.

Multiplicative functions

Teilersumme | Fonction diviseur | 약수 함수 | 約数関数 | פונקציית מחלקים

$\sigma_{0}(12)$	$= 1^0 + 2^0 + 3^0 + 4^0 + 6^0 + 12^0$
	$= 1 + 1 + 1 + 1 + 1 + 1 = 6.$

$\sigma_{1}(12)$	$= 1^1 + 2^1 + 3^1 + 4^1 + 6^1 + 12^1$
	$= 1 + 2 + 3 + 4 + 6 + 12 = 28.$

$\tau(24)$	$= \prod_{i=1}^{2} (a_i+1)$
	$= (3 + 1)(1 + 1) = 4 \times 2 = 8.$