Minkowski's second theorem

An asymptotic expression for p(n) is given by

$p(n)\sim {\frac {1}{4n{\sqrt {3}}}}\exp \left({\pi {\sqrt {\frac {2n}{3}}}}\right){\mbox{ as }}n\rightarrow \infty .$

This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920. Considering p(1000), the asymptotic formula gives about 2.4402 × 10³¹, reasonably close to the exact answer given above (1.415% larger than the true value).

Hardy and Ramanujan obtained an asymptotic expansion with this approximation as the first term:

$p(n)={\frac {1}{2{\sqrt {2}}}}\sum _{k=1}^{v}{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\exp \left({\pi {\sqrt {\frac {2}{3}}}{\frac {\sqrt {n-{\frac {1}{24}}}}{k}}}\right)$

where

$A_{k}(n)=\sum _{0\,\leq \,m\,<\,k;\;(m,\,k)\,=\,1}e^{\pi i\left[s(m,\,k)\;-\;{\frac {1}{k}}2nm\right]}.$

Here, the notation (m, n) = 1 implies that the sum should occur only over the values of m that are relatively prime to n. The function s(m, k) is a Dedekind sum.

The error after v terms is of the order of the next term, and v may be taken to be of the order of ${\sqrt {n}}$. As an example, Hardy and Ramanujan showed that p(200) is the nearest integer to the sum of the first v=5 terms of the series.

In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan's results by providing a convergent series expression for p(n). It is

$p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\left({{\frac {1}{\sqrt {n-{\frac {1}{24}}}}}\sinh \left[{{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}\left(n-{\frac {1}{24}}\right)}}}\right]}\right).$

It can be shown that the derivative part of the sum can be simplified. The proof of Rademacher's formula involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function in a central way.

It may be shown that the k-th term of Rademacher's series is of the order

$\exp \left(\pi {\sqrt {\frac {2}{3}}}{\frac {\sqrt {n}}{k}}\right),$

so that the first term gives the Hardy–Ramanujan asymptotic approximation.