The divisor function for n an integer is defined as the sum of the kth powers of the (integer) divisors of n,

It is implemented in Mathematica as DivisorSigma[k, n].

The divisor function can also be generalized to Gaussian integers. The definition requires some care since in principle, there is ambiguity as to which of the four associates is chosen for each divisor. Spira (1961) defines the sum of divisors of a complex number z by factoring z 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 Mathematica as DivisorSigma[1, z, GaussianIntegers -> True]. The following table gives for small nonnegative values of a and b.

The notations d(n) (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 n. Rather surprisingly, the number of factors of the polynomial are also given by d(n). 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 n.

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

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 N = 12 divisors in all. Therefore,

The following table summarized the first few values of for small k and n = 1, 2, ....

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

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

For a given ,

Summing over all ,

so . Splitting a and b into prime factors,

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

For N, therefore,

(Berndt 1985).

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

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

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

The function has the series expansion

	(21)

(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).

The divisor function also satisfies the inequality

where is the Euler-Mascheroni constant (Robin 1984, Erdos 1989).

Gronwall's theorem states that

where is the Euler-Mascheroni constant (Hardy and Wright 1979, p. 266). is a power of 2 iff n = 1 or n is a product of distinct Mersenne primes (Sierpinski 1958/59, Sivaramakrishnan 1989, Kaplansky 1999). The first few such n are 1, 3, 7, 21, 31, 93, 127, 217, 381, 651, 889, 2667, ... (Sloane's 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, ... (Sloane's A048947).

Curious identities derived using modular form theory are given by

(Apostol 1997, p. 140), together with

	(30)

(M. Trott).

The divisor function is odd iff n 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 d(n) is prime whenever is (Honsberger 1991). Factorizations of for prime p are given by Sorli.

In 1838, Dirichlet showed that the average number of divisors of all numbers from 1 to n 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 a > 1 are

For a = 1,

(Hardy and Wright 1979, p. 266).

 

Dirichlet Divisor Problem, Distinct Prime Factors, Divisor, Divisor Product, Even Divisor Function, Factor, Fermat's Divisor Problem, Greatest Prime Factor, Gronwall's Theorem, Highly Composite Number, Least Prime Factor, Multiply Perfect Number, Odd Divisor Function, Ore's Conjecture, Perfect Number, Prime Factor, Refactorable Number, Restricted Divisor Function, Robin's Theorem, Silverman Constant, Sociable Numbers, Sum of Squares Function, Superabundant Number, Tau Function, Totient Function, Totient Valence Function, Twin Peaks, Unitary Divisor Function