# 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 σx(n) is defined as the sum of the xth powers of the positive divisors of n, or

\sigmax(n)=\sumd|ndx.

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

nDivisorsσ0(n)σ1(n)Aliquot sum
11110
21,2231
31,3241
41,2,4373
51,5261
61,2,3,64126
71,7281
81,2,4,84157
91,3,93134
101,2,5,104188
111,112121
121,2,3,4,6,1262816
131,132141
141,2,7,1442410
151,3,5,154249

## 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(pn\#)=2n

where pn# 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

n=

 r \prod i=1
 ai p i
where

r=\omega(n)

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

\sigmax(n)=

 r \prod i=1
 (ai+1)x p -1 i
 x-1 p i

which is equivalent to the useful formula:

\sigmax(n)=

 r \prod i=1
 ai \sum j=0
 jx p i

=

 r \prod i=1

(1+

 x p i

+

 2x p i

+...+

 aix p i

).

It follows (by setting x = 0) that

d(n)

is:
 r d(n)=\prod i=1

(ai+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

(ai+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

\sigmax(n)=\sum

 n \mu=1

\mux-1

 \mu \sum \cos \nu=1
 2\pi\nun \mu
without an explicit reference to the divisors of

n

.[1]

## Series relations

Two Dirichlet series involving the divisor function are:

 infty \sum n=1
 \sigmaa(n) ns

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

and
 infty \sum n=1
 \sigmaa(n)\sigmab(n) = ns
 \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\sigmaa(n)=

 infty \sum n=1
 naqn 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

\limsupn\toinfty

 logd(n) logn/loglogn

=log2.

On the other hand, clearly

\liminfn\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,\sumn\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:

\limsupn → 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)\leHn+

 Hn ln(H n)e
for every natural number n, where

Hn

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.
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