Divisor Function

The divisor function for a positive integer is defined as the sum of the th powers of the (positive integer) divisors of ,

It is implemented in the Wolfram Language as DivisorSigma[k, n].

The notations (Hardy and Wright 1979, p. 239), (Ore 1988, p. 86), and (Burton 1989, p. 128) are sometimes used for , which gives the number of divisors of . Rather surprisingly, the number of distinct irreducible polynomials in the polynomial factorization of over the rational numbers, with and treated as indeterminates, is also given by . The values of can be found as the inverse Möbius transform of 1, 1, 1, ... (Sloane and Plouffe 1995, p. 22). Heath-Brown (1984) proved that infinitely often. The numbers having the incrementally largest number of divisors are called highly composite numbers. The function satisfies the identities

where the are distinct primes and is the prime factorization of a number .

The divisor function is odd iff is a square number.

The function that gives the sum of the divisors of is commonly written without the subscript, i.e., .

Li (2026) considered the number of ordered pairs

where and are positive integers. Guy (1994) records the question of whether for some constant . This is problem 1061 in the Erdős problems collection (Bloom 2026). Li (2026) instead reported

for every fixed . Large language models, primarily ChatGPT, contributed to technical development of the proof, and Aristotle produced an accompanying Lean formalization under the author's direction. As of Sep. 21, 2026, independent specialist review had not been reported, and VibeMathed had not independently compiled the formalization (VibeMathed 2026).

As an illustrative example of computing , consider the number 140, which has divisors , 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140, for a total of divisors in all. Therefore,

The following table summarizes the first few values of for small and , 2, ....

The sum of the divisors of excluding itself (i.e., the proper divisors of ) is called the restricted divisor function and is denoted . The first few values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... (OEIS A001065).

The sum of divisors can be found as follows. Let with and . For any divisor of , , where is a divisor of and is a divisor of . The divisors of are 1, , , ..., and . The divisors of are 1, , , ..., . The sums of the divisors are then

For a given ,

Summing over all ,

so . Splitting and into prime factors,

For a prime power , the divisors are 1, , , ..., , so

For , therefore,

(Berndt 1985).

For the special case of a prime, (16) simplifies to

Similarly, for a power of two, (16) simplifies to

The identities (14) and (16) can be generalized to

when for all .

Sums involving the divisor function are given by

for ,

for , and more generally,

for and (Hardy and Wright 1979, p. 250).

A generating function for is given by the Lambert series

where is the order-zero q-polygamma function and .

The function has the series expansion

(Hardy 1999). Ramanujan gave the beautiful formula

where is the zeta function and (Wilson 1923), which was used by Ingham in a proof of the prime number theorem (Hardy 1999, pp. 59-60). This gives the special case

(Hardy 1999, p. 59).

Gronwall's theorem states that

where is the Euler-Mascheroni constant (Hardy and Wright 1979, p. 266; Robin 1984). Robin (1984, Theorem 2) proved the explicit inequality

for , with equality at , giving

The Riemann hypothesis is equivalent to the statement that

for all (Robin 1984, Theorem 1).

is a power of 2 iff or is a product of distinct Mersenne primes (Sierpiński 1963, Sivaramakrishnan 1989, Kaplansky 1999). The first few such are 1, 3, 7, 21, 31, 93, 127, 217, 381, 651, 889, 2667, ... (OEIS A046528), and the powers of 2 these correspond to are 0, 2, 3, 5, 5, 7, 7, 8, 9, 10, 10, 12, 12, 13, 14, ... (OEIS A048947).

Curious identities derived using modular form theory are given by

(Apostol 1997, p. 140), together with

(M. Trott, pers. comm.).

The divisor function (and, in fact, for ) is odd iff is a square number or twice a square number. The divisor function satisfies the congruence

for all primes and no composite numbers with the exception of 4, 6, and 22 (Subbarao 1974).

The number of divisors is prime whenever itself is prime (Honsberger 1991). Factorizations of for prime are given by Sorli.

Dirichlet (1838) showed that the average number of divisors of all numbers from 1 to is asymptotic to

(Conway and Guy 1996; Hardy 1999, p. 55; Havil 2003, pp. 112-113), as illustrated above, where the thin solid curve plots the actual values and the thick dashed curve plots the asymptotic function. This is related to the Dirichlet divisor problem, which seeks to find the "best" coefficient in

(Hardy and Wright 1979, p. 264).

The summatory functions for with are

For ,

(Hardy and Wright 1979, p. 266).

The divisor function can also be generalized to Gaussian integers. The definition requires some care since each divisor has four multiples by the units , , , and . The Wolfram Language function Divisors[z] chooses the unique multiple satisfying and from each group of four. Reapplying this first-quadrant convention after multiplication does not give a multiplicative function.

By contrast, Spira (1961) defines the sum of divisors of a complex number by factoring into a product of powers of distinct Gaussian primes,

where is a unit and each lies in the first quadrant of the complex plane, and then writing

This makes a multiplicative function and also gives . This extension is implemented in the Wolfram Language as DivisorSigma[1, z, GaussianIntegers -> True].

For example, let , and let denote the list returned by Divisors[q]. Direct computation gives

Thus satisfies the first identity, but is not a perfect Gaussian integer in Spira's multiplicative sense (Sanchez Solovieva 2026).

The following table gives for small nonnegative values of and .

The number of distinct prime factors is computed in the Wolfram Language using PrimeNu.