# 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.

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

The **aliquot sum** *s*(*n*) of *n* is the sum of the proper divisors (that is, the divisors excluding *n* itself, ), and equals σ_{1}(*n*) − *n*; the aliquot sequence of *n* is formed by repeatedly applying the aliquot sum function.

because by definition, the factors of a prime number are 1 and itself. Also, where *p _{n}*# denotes the primorial,

The divisor function is multiplicative,^{[why?]} but not completely multiplicative:

where *r* = *ω*(*n*) 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:

^{[4]}

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.

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

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.

In little-o notation, the divisor function satisfies the inequality:^{[12]}^{[13]}

In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality:^{[14]}^{[15]}

The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by: ^{[16]}

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:

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the 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).

A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that: