# 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.

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## Contents

## Definition

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

The notations *d*(*n*) and (the tau function) are also used to denote σ_{0}(*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.

For example, σ_{0}(12) may be considered as the sum of the zeroth powers of the divisors of 12:

while σ_{1}(12) is equal to the sum of the divisors' first powers:

For a prime number *p*,

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

where *r* is the number of distinct prime factors of *n*, *p _{i}* is the

*i*

^{th}prime factor, and

*a*is the maximum power of

_{i}*p*by which

_{i}*n*is divisible, then we have

which is equivalent to the useful formula:

An equation for calculating is

For example, if *n* is 24, there are two prime factors (*p _{1}* is 2;

*p*is 3); noting that 24 is the product of 2

_{2}^{3}×3

^{1},

*a*is 3 and

_{1}*a*is 1. Thus we can calculate as so:

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

We also note .
This function is the one used to recognize perfect numbers which are the *n* for which . 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

Then

## Properties

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:

and

A Lambert series involving the divisor function is:

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.

## Inequalities

For the number of divisors function,

- , for n > 12.

Another bound on the number of divisors is

- , for n ≥ 3.

For the sum of divisors function,

- , for n > 12.

A pair of inequalities combining the divisor function and the φ function are:

- , for n > 1.

## Approximate growth rate

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

where is Euler's constant. This result is Gronwall's Theorem, published in 1913.

Interestingly, in 1984 Guy Robin proved that

- for
*n*> 5,040

is true if and only if the Riemann hypothesis is true. The largest known value that violates the inequality is *n*=5,040. If the Riemann hypothesis is true, there are no others. 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

for every natural number *n*, where is the *n*th harmonic number.

## See also

- Euler's totient function (Euler's phi function)
- Riemann zeta function

## 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.