A fundamental problem studies the number of ways 
can be written as a sum
of positive integers 
, that is, the number of solutions of
.
: positive integerThe number of summands is unrestricted, repetition is allowed, and the order of
the summands is not taken into account. The corresponding unrestricted
partition function is denoted by 
, and the summands are
called parts; see §26.9(i). For example,
because there are exactly seven partitions of 5:
.
The number of partitions of into at most 
parts is denoted by
; again see §26.9(i).
Euler introduced the reciprocal of the infinite product
,
: Euler’s reciprocal function
: positive integer and
: real number
as a generating function for the function defined in
§27.14(i):
: Euler’s reciprocal function,
: total number of partitions of ,
: positive integer and
: real number
with 
. Euler’s pentagonal number theorem states
that
: Euler’s reciprocal function,
: pentagonal numbers,
: positive integer and
: real number
where the exponents 1, 2, 5, 7, 12, 15, 
are the
pentagonal numbers, defined by
.
: pentagonal numbers: positive integerMultiplying the power series for with that for
and equating coefficients, we obtain the recursion formula
: total number of partitions of ,
: pentagonal numbers,
: positive integer and
: positive integer
where 
is defined to be 0 if 
. Logarithmic
differentiation of the generating function leads to another
recursion:
: sum of powers of divisors of ,
: total number of partitions of ,
: positive integer and
: positive integer
where 
is defined by (27.2.10) with
.
These recursions can be used to calculate , which grows
very rapidly. For example, 
= 1905 69292, and
. For large
: base of exponential function,
: total number of partitions of ,
: asymptotic equality and
: positive integer
where 
(Hardy and Ramanujan (1918)).
Rademacher (1938) derives a convergent series that also provides an
asymptotic expansion for :
![\mathop{p\/}\nolimits\!\left(n\right)=\frac{1}{\pi\sqrt{2}}\sum_{{k=1}}^{%
\infty}\sqrt{k}A_{k}(n)\*\left[\frac{d}{dt}\frac{\mathop{\sinh\/}\nolimits\!%
\left(\ifrac{K\sqrt{t}}{k}\right)}{\sqrt{t}}\right]_{{t=n-(1/24)}},](./27/14/E9.png)
: derivative of 
with respect to ,
: hyperbolic sine function,
: total number of partitions of ,
: positive integer,
: positive integer and
: coefficients
where
: Dedekind sum,
: exponential function,
: positive integer,
: positive integer and
: coefficients
and 
is a Dedekind sum given by
: Dedekind sum
: floor of and
: positive integer
Dedekind sums occur in the transformation theory of the Dedekind modular
function 
, defined by
.
: Dedekind’s eta function (or Dedekind modular function): base of exponential function,
: imaginary part and
: positive integer
This is related to the function in (27.14.2) by
: Dedekind’s eta function (or Dedekind modular function),
: Euler’s reciprocal function,
: base of exponential function and
: imaginary part
satisfies the following functional equation: if
are integers with 
and 
, then
: Dedekind’s eta function (or Dedekind modular function),
: imaginary part,
: integer,
: integer,
: integer and
: integer
where 
and 
is
given by (27.14.11).
Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity
: Euler’s reciprocal function,
: total number of partitions of ,
: positive integer and
: real number
implies 
. Ramanujan also found that
and
for all . After decades of nearly
fruitless searching for further congruences of this type, it was believed that
no others existed, until it was shown in Ono (2000) that there are
infinitely many.
Ono proved that for every prime 
there are integers and
such that 
for all . For example,
.
The discriminant function 
is defined by
,
: discriminant function: Dedekind’s eta function (or Dedekind modular function) and
: imaginary part
and satisfies the functional equation
: discriminant function and
: imaginary part
if are integers with and .
The 24th power of in (27.14.12) with
is an infinite product that generates a power series in
with integer coefficients called Ramanujan’s tau function
:
.
: Ramanujan’s tau function: positive integer and
: real number
The tau function is multiplicative and satisfies the more general relation:
.
: Ramanujan’s tau function,
: positive integer and
: positive integer
Lehmer (1947) conjectures that is never 0 and
verifies this for all 
by studying various congruences
satisfied by , for example:
: Ramanujan’s tau function,
: sum of powers of divisors of and
: positive integer
