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
![sigma_1(n)=1/6npi^2[(1+((-1)^n)/(2^2))+(2cos(2/3npi))/(3^2)+(2cos(1/2npi))/(4^2)+(2[cos(2/5npi)+cos(4/5npi)])/(5^2)+...]](/images/equations/DivisorFunction/NumberedEquation12.svg) |
(26)
|
(Hardy 1999). Ramanujan gave the beautiful formula
 |
(27)
|
where 
is the zeta function and ![R[s],R[s-a],R[s-b],R[s-a-b]>1](/images/equations/DivisorFunction/Inline117.svg)
(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
![sum_(n=1)^infty([d(n)]^2)/(n^s)=([zeta(s)]^4)/(zeta(2s))](/images/equations/DivisorFunction/NumberedEquation14.svg) |
(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 |  |  |  |  |  |  |  |
,"
"Analogs with 
,
,"
"Solutions of 
,"
and "Solutions of 
." §B11, B12, B13 and
B15 in