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Wikipedia / Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). 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; these are treated separately in the article Ramanujan's sum.

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 σz(n), for a real or complex number z, is defined as the sum of the zth powers of the positive divisors of n. It can be expressed in sigma notation as

is shorthand for "d divides n". The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function (). When z is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is the same as σ1(n) ().

The aliquot sum s(n) 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:

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

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

σ−1(n) is sometimes called the abundancy index of n, and we have:

Table of values

The cases x = 2 to 5 are listed in through , x = 6 to 24 are listed in through .

Properties

Formulas at prime powers

For a prime number p,

.

, then

General formula

, where r=ω(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:

which, when x0, is equivalent to:

When x= is:

Proof: The proof is combinatorial in nature. Distributing (expanding) the product

.

. These are the same summands obtained when expanding

is multiplicative are disjoint. Then

Other properties and identities

Euler proved the remarkable recurrence:

are consecutive pairs of generalized pentagonal numbers (, starting at offset 1). Indeed, Euler proved this by logarithmic differentiation of the identity in his pentagonal number theorem.

For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n is odd if and only if n is a square or twice a square.

We also note s(n) = σ(n)−n. Here s(n) denotes the sum of the proper divisors of n, that is, the divisors of n excluding n itself. This function is used to recognize perfect numbers, which are the n such that s(n) =n. If s(n) > n, then n is an abundant number, and if s(n) < n, then n is a deficient number.

If , which makes n almost-perfect.

, let

  1. .

Then

and

is Euler's totient function.

Then, the roots of

express p and q in terms of σ(n) and φ(n) only, requiring no knowledge of n, as

Also, knowing allows an easy recovery of p and q.

In 1984, Roger Heath-Brown proved that the equality

is true for infinitely many values of , see .

Dirichlet convolutions

By Möbius inversion

Series relations

Two Dirichlet series involving the divisor function are:

is the Riemann zeta function. The series for d(n)=0(n) gives:

and a Ramanujan identity

which is a special case of the Rankin–Selberg convolution.

A Lambert series involving the divisor function is:

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

, there is an explicit series representation with Ramanujan sums as :

:

Growth rate

In little-o notation, the divisor function satisfies the inequality:

More precisely, Severin Wigert showed that:

taking the primorials since

On the other hand, since there are infinitely many prime numbers,

In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:

is Euler's gamma constant in this formula is known as Dirichlet's divisor problem.

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

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 . His proof uses Mertens' third theorem, which says that:

where p denotes a prime. Grönwall also showed that

is the Riemann zeta function.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, Robin's inequality

  1. (where γ is the Euler–Mascheroni constant)

holds for all sufficiently large n . The largest known value that violates the inequality is n=5040. In 1984, Guy Robin proved that the inequality is true for all n > 5040 if and only if the Riemann hypothesis is true . This is Robin's theorem and the inequality became known after him. Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of n that violate the inequality, and it is known that the smallest such n > 5040 must be superabundant . It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for n divisible by the fifth power of a prime .

Robin also proved, unconditionally, that the inequality:

holds for all n ≥ 3.

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 is the nth harmonic number, .

See also

Notes

References