Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the 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.

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 where

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[1][2] (OEIS: A000005).

The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, OEIS: A001065), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.

Also, where pn# denotes the primorial, since n prime factors allow a sequence of binary selection (

However, these are not in general the smallest numbers whose number of divisors is a power of two; instead, the smallest such number may be obtained by multiplying together the first n Fermi–Dirac primes, prime powers whose exponent is a power of two.

distinctively correspond to a divisor a of m and a divisor b of n), but not completely multiplicative: The consequence of this is that, if we write 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: [5] which, when x ≠ 0, is equivalent to the useful formula: [5] When x = 0,

is: [5] This result can be directly deduced from the fact that all divisors of

Euler proved the remarkable recurrence:[6][7][8] where

are consecutive pairs of generalized pentagonal numbers (OEIS: A001318, starting at offset 1).

Indeed, Euler proved this by logarithmic differentiation of the identity in his pentagonal number theorem.

Then, the roots of express p and q in terms of σ(n) and φ(n) only, requiring no knowledge of n or

In 1984, Roger Heath-Brown proved that the equality is true for infinitely many values of n, see OEIS: A005237.

Two Dirichlet series involving the divisor function are: [10] where

The series for d(n) = σ0(n) gives: [10] and a Ramanujan identity[11] which is a special case of the Rankin–Selberg convolution.

A Lambert series involving the divisor function is: [12] 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.

, there is an explicit series representation with Ramanujan sums

: In little-o notation, the divisor function satisfies the inequality:[14][15] More precisely, Severin Wigert showed that:[15] On the other hand, since there are infinitely many prime numbers,[15] In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:[16][17] where

The behaviour of the sigma function is irregular.

The asymptotic growth rate of the sigma function can be expressed by: [18] where lim sup is the limit superior.

His proof uses Mertens' third theorem, which says that: where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, Robin's inequality holds for all sufficiently large n (Ramanujan 1997).

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: 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 > 1, where

is the nth harmonic number, (Lagarias 2002).