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

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Divisor Functions." §24.3.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 827, 1972.

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 140, 1997.

Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, p. 94, 1985.

Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 260-261, 1996.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 279-325, 2005.

Dirichlet, G. L. "Sur l'usage des séries infinies dans la théorie des nombres." J. reine angew. Math. 18, 259-274, 1838.

Guy, R. K. "Solutions of ," "Analogs with , ," "Solutions of ," and "Solutions of ." §B11, B12, B13 and B15 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 67-70, 1994.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 55 and 141, 1999.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 354-355, 1979.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.

Heath-Brown, D. R. "A Parity Problem from Sieve Theory." Mathematika 29, 1-6, 1982.

Heath-Brown, D. R. "The Divisor Function at Consecutive Integers." Mathematika 31, 141-149, 1984.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 250-251, 1991.

Kaplansky, I. "The First Two Chapters of Dickson's History." Unpublished manuscript, Apr. 1999.

Nagell, T. Introduction to Number Theory. New York: Wiley, pp. 26-27, 1951.

Ore, Ø. Number Theory and Its History. New York: Dover, 1988.

Robin, G. "Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann." J. Math. Pures Appl. 63, 187-213, 1984.

Sierpiński, W. "Sur les nombres dont la somme de diviseurs est une puissance du nombre 2." Calcutta Math. Soc. Golden Jubilee Commemoration 1958/59, Part I. Calcutta: Calcutta Math. Soc., pp. 7-9, 1963.

Sloane, N. J. A. Sequences A000005, A000203, A001065, A001157, A001158, A046528, and A048947 in "The On-Line Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

Sivaramakrishnan, R. Classical Theory of Arithmetic Functions. New York: Dekker, 1989.

Sorli, R. "Factorization Tables." https://www-staff.maths.uts.edu.au/~rons/fact/fact.htm.

Spira, R. "The Complex Sum of Divisors." Amer. Math. Monthly 68, 120-124, 1961.

Subbarao, M. V. "On Two Congruences for Primality." Pacific J. Math. 52, 261-268, 1974.

Wilson, B. M. "Proofs of Some Formulae Enunciated by Ramanujan." Proc. London Math. Soc. 21, 235-255, 1923.