Divisor Function

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

(1)

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 factors of the polynomial are 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

			(2)
			(3)

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., .

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,

			(4)
			(5)
			(6)
			(7)

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

	OEIS	for , 2, ...
0	A000005	1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ...
1	A000203	1, 3, 4, 7, 6, 12, 8, 15, 13, 18, ...
2	A001157	1, 5, 10, 21, 26, 50, 50, 85, 91, 130, ...
3	A001158	1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, ...

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

			(8)
			(9)

For a given ,

(10)

Summing over all ,

(11)

so . Splitting and into prime factors,

(12)

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

(13)

For , therefore,

(14)

(Berndt 1985).

For the special case of a prime, (14) simplifies to

(15)

Similarly, for a power of two, (14) simplifies to

(16)

The identities (◇) and (◇) can be generalized to

			(17)
			(18)

Sums involving the divisor function are given by

(19)

for ,

(20)

for , and more generally,

(21)

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

A generating function for is given by the Lambert series

			(22)
			(23)
			(24)
			(25)

where is a q-polygamma function.

The function has the series expansion

(26)

(Hardy 1999). Ramanujan gave the beautiful formula

(27)

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

(28)

(Hardy 1999, p. 59).

Gronwall's theorem states that

(29)

where is the Euler-Mascheroni constant (Hardy and Wright 1979, p. 266; Robin 1984). This can be written as an explicit inequality as

(30)

where is the Euler-Mascheroni constant and where equality holds for , giving

			(31)
			(32)

(Robin 1984, Theorem 2). In fact, the constant term can be dropped if the Riemann hypothesis holds, since the Riemann hypothesis is equivalent to the statement that

(33)

for all (Robin 1984, Theorem 1).

is a power of 2 iff or is a product of distinct Mersenne primes (Sierpiński 1958/59, 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

(34)

(35)

(Apostol 1997, p. 140), together with

(36)

(37)

(38)

(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

(39)

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.

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

(40)

(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

(41)

(Hardy and Wright 1979, p. 264).

The summatory functions for with are

(42)

For ,

(43)

(Hardy and Wright 1979, p. 266).

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 by factoring into a product of powers of distinct Gaussian primes,

(44)

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

(45)

This makes a multiplicative function and also gives . This extension is implemented in the Wolfram Language as DivisorSigma[1, z, GaussianIntegers -> True]. The following table gives for small nonnegative values of and .

	0	1	2	3	4	5	6
1	1
2
3	4
4
5
6

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