Divisor function explained

In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to 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 sum of positive divisors 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|nd^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, so σ(n) is equivalent to σ₁(n) . The aliquot sum of n is the sum of the proper divisors (that is, the divisors excluding n itself ), and equals σ₁(n) - n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.

Example

For example, σ₀(12) is the number of the divisors of 12:

\sigma₀(12)

=1⁰+2⁰+3⁰+4⁰+6⁰+12⁰

=1+1+1+1+1+1=6.

while σ₁(12) is the sum of all the divisors:

\sigma₁(12)

=1¹+2¹+3¹+4¹+6¹+12¹

=1+2+3+4+6+12=28.

and the aliquot sum s(12) of proper divisors is:

s(12)

=1¹+2¹+3¹+4¹+6¹

=1+2+3+4+6=16.

Table of values

n	Divisors	σ₀(n)	σ₁(n)	Aliquot sum
1	1	1	1	0
2	1,2	2	3	1
3	1,3	2	4	1
4	1,2,4	3	7	3
5	1,5	2	6	1
6	1,2,3,6	4	12	6
7	1,7	2	8	1
8	1,2,4,8	4	15	7
9	1,3,9	3	13	4
10	1,2,5,10	4	18	8
11	1,11	2	12	1
12	1,2,3,4,6,12	6	28	16
13	1,13	2	14	1
14	1,2,7,14	4	24	10
15	1,3,5,15	4	24	9

Properties

For a prime number p,

d(p)=2

d(pⁿ⁾=n+1

\sigma(p)=p+1

because by definition, the factors of a prime number are 1 and itself. Also

d(p_n\#)=2ⁿ

where p_n# denotes the primorial.

Clearly, 1 < d(n) < n and σ(n) > n for all n > 2.

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

	r
\prod
	i=1

	a_i
p
	i

where

r=\omega(n)

is the number of distinct prime factors of n, p_i is the ith prime factor, and a_i is the maximum power of p_i by which n is divisible, then we have

\sigma_x(n)=

	r
\prod
	i=1

	(a_i+1)x
p		-1
	i

	x-1
p
	i

which is equivalent to the useful formula:

\sigma_x(n)=

	r
\prod
	i=1

	a_i
\sum
	j=0

	jx
p
	i

	r
\prod
	i=1

(1+

	x
p
	i

	2x
p
	i

+...+

	a_ix
p
	i

It follows (by setting x = 0) that

d(n)

is:

	r
d(n)=\prod
	i=1

(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

d(24)

as so:

d(24)

=

2
\prod
i=1

(a_i+1)

=(3+1)(1+1)=4 x 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

. Here

s(n)

denotes the sum of the proper divisors of n, i.e. the divisors of n excluding n itself.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 expansion

The divisor function can be written as a finite trigonometric series

\sigma_x(n)=\sum

	n

	\mu=1

\mu^x-1

	\mu
\sum		\cos
	\nu=1

	2\pi\nun
	\mu

without an explicit reference to the divisors of

.^[1]

Series relations

Two Dirichlet series involving the divisor function are:

	infty
\sum
	n=1

	\sigma_a(n)
	n^s

=\zeta(s)\zeta(s-a)

and

	infty
\sum
	n=1

	\sigma_a(n)\sigma_b(n)	=
	n^s

	\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)
	\zeta(2s-a-b)

A Lambert series involving the divisor function is:

	infty
\sum
	n=1

qⁿ\sigma_a(n)=

	infty
\sum
	n=1

	n^aqⁿ
	1-qⁿ

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

In little-o notation, the divisor function satisfies the inequality (see page 296 of Apostol’s book)

forall\epsilon>0, d(n)=o(n^\epsilon).

More precisely, Severin Wigert showed that

\limsup_n\toinfty

	logd(n)
	logn/loglogn

=log2.

On the other hand, clearly

\liminf_n\toinftyd(n)=2.

In Big-O notation, Dirichlet showed that the average order of the divisor function satisfies the following inequality (see Theorem 3.3 of Apostol’s book)

forallx\geq1,\sum_n\leqd(n)=xlogx+(2\gamma-1)x+O(\sqrt{x}),

where

\gamma

is Euler's constant. Improving the bound

O(\sqrt{x})

in this formula is known as Dirichlet's divisor problem

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

\limsup_{n → infty}

	\sigma(n)
	nloglogn

=e^\gamma,

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913.

In 1984 Guy Robin proved that

\sigma(n)<e^\gammanloglogn

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, the smallest of which must be a superabundant number.

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

\sigma(n)\leH_n+

	H_n
ln(H
	n)e

for every natural number n, where

H_n

is the nth harmonic number.

References

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.

[w:de:Teilersumme#Teilersumme als endliche Reihe]

\sigma₀(12)	=1⁰+2⁰+3⁰+4⁰+6⁰+12⁰
	=1+1+1+1+1+1=6.