Divisor function - Alchetron, The Free Social Encyclopedia

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. 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.

Definition

The sum of positive divisors function σ_x(n), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as

σ x ( n ) = ∑ d ∣ n d x ,

where d ∣ n 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 ( A000005). 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 the same as σ₁(n) ( A000203).

The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, A001065), 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:

σ 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 the sum of all the divisors:

σ 1 ( 12 ) = 1 1 + 2 1 + 3 1 + 4 1 + 6 1 + 12 1 = 1 + 2 + 3 + 4 + 6 + 12 = 28 ,

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

s ( 12 ) = 1 1 + 2 1 + 3 1 + 4 1 + 6 1 = 1 + 2 + 3 + 4 + 6 = 16.

Table of values

The cases x = 2 to 5 are listed in A001157 − A001160, x = 6 to 24 are listed in A013954 − A013972.

Properties

For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and σ 0 ( n ) is even; for a square integer, one divisor (namely n ) is not paired with a distinct divisor and σ 0 ( n ) is odd. Similarly, the number σ 1 ( n ) is odd if and only if n is a square or twice a square.

For a prime number p,

σ 0 ( p ) = 2 σ 0 ( p n ) = n + 1 σ 1 ( p ) = p + 1

because by definition, the factors of a prime number are 1 and itself. Also, where p_n# denotes the primorial,

σ 0 ( p n # ) = 2 n

since n prime factors allow a sequence of binary selection ( p i or 1) from n terms for each proper divisor formed.

Clearly, 1 < σ 0 ( 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 = ∏ i = 1 r p i a i

where r = ω(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

σ x ( n ) = ∏ i = 1 r p i ( a i + 1 ) x − 1 p i x − 1

which is equivalent to the useful formula:

σ x ( n ) = ∏ i = 1 r ∑ j = 0 a i p i j x = ∏ i = 1 r ( 1 + p i x + p i 2 x + ⋯ + p i a i x ) .

It follows (by setting x = 0) that d(n) is:

σ 0 ( n ) = ∏ 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 σ 0 ( 24 ) as so:

σ 0 ( 24 ) = ∏ i = 1 2 ( a i + 1 ) = ( 3 + 1 ) ( 1 + 1 ) = 4 ⋅ 2 = 8.

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

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 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.

If n is a power of 2, for example, n = 2 k , then σ ( n ) = 2 ⋅ 2 k − 1 = 2 n − 1 , and s(n) = n - 1, which makes n almost-perfect.

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

n = p q .

Then

σ ( n ) = ( p + 1 ) ( q + 1 ) = n + 1 + ( p + q ) , φ ( n ) = ( p − 1 ) ( q − 1 ) = n + 1 − ( p + q ) ,

and

n + 1 = ( σ ( n ) + φ ( n ) ) / 2 , p + q = ( σ ( n ) − φ ( n ) ) / 2 ,

where φ(n) is Euler's totient function.

Then, the roots of:

( x − p ) ( x − q ) = x 2 − ( p + q ) x + n = x 2 − [ ( σ ( n ) − φ ( n ) ) / 2 ] x + [ ( σ ( n ) + φ ( n ) ) / 2 − 1 ] = 0

allow us to express p and q in terms of σ(n) and φ(n) only, without even knowing n or p+q, as:

p = ( σ ( n ) − φ ( n ) ) / 4 − [ ( σ ( n ) − φ ( n ) ) / 4 ] 2 − [ ( σ ( n ) + φ ( n ) ) / 2 − 1 ] , q = ( σ ( n ) − φ ( n ) ) / 4 + [ ( σ ( n ) − φ ( n ) ) / 4 ] 2 − [ ( σ ( n ) + φ ( n ) ) / 2 − 1 ] .

Also, knowing n and either σ(n) or φ(n) (or knowing p+q and either σ(n) or φ(n)) allows us to easily find p and q.

In 1984, Roger Heath-Brown proved that the equality

σ 0 ( n ) = σ 0 ( n + 1 )

is true for an infinity of values of n, see A005237.

Series relations

Two Dirichlet series involving the divisor function are:

∑ n = 1 ∞ σ a ( n ) n s = ζ ( s ) ζ ( s − a ) for s > 1 , s > a + 1 ,

which for d(n) = σ₀(n) gives

∑ n = 1 ∞ d ( n ) n s = ζ 2 ( s ) for s > 1 ,

and

∑ n = 1 ∞ σ a ( n ) σ b ( n ) n s = ζ ( s ) ζ ( s − a ) ζ ( s − b ) ζ ( s − a − b ) ζ ( 2 s − a − b ) .

A Lambert series involving the divisor function is:

∑ n = 1 ∞ q n σ a ( n ) = ∑ n = 1 ∞ 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.

Growth rate

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

for all ϵ > 0 , d ( n ) = o ( n ϵ ) .

More precisely, Severin Wigert showed that

lim sup n → ∞ log ⁡ d ( n ) log ⁡ n / log ⁡ log ⁡ n = log ⁡ 2.

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

lim inf n → ∞ d ( n ) = 2.

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

for all x ≥ 1 , ∑ n ≤ x d ( n ) = x log ⁡ x + ( 2 γ − 1 ) x + O ( x ) ,

where γ is Euler's gamma constant. Improving the bound O ( x ) 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:

lim sup n → ∞ σ ( n ) n log ⁡ log ⁡ n = e γ ,

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 (Grönwall 1913). His proof uses Mertens' 3rd theorem, which says that

lim n → ∞ 1 log ⁡ n ∏ p ≤ n p p − 1 = e γ ,

where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality:

σ ( n ) < e γ n log ⁡ log ⁡ n (Robin's inequality)

holds for all sufficiently large n (Ramanujan 1997). 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 (Robin 1984). 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 (Akbary & Friggstad 2009). 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 (Choie et al. 2007).

Robin also proved, unconditionally, that the inequality

σ ( n ) < e γ n log ⁡ log ⁡ n + 0.6483 n log ⁡ log ⁡ n

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

σ ( n ) < H n + ln ⁡ ( H n ) e H n

for every natural number n > 1, where H n is the nth harmonic number, (Lagarias 2002).

References

Divisor function Wikipedia

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