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Adding the Rademacher's formula for partition function. |
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:<math>p(n) \sim \frac {1} {4n\sqrt3} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right)</math> as <math>n \to \infty</math>
In 1937, [[Hans Rademacher]] had find out a way to represent the partition function <math>p(n)</math> by the [[convergent series]].
<math display="block">p(n) = \frac{1}{\pi \sqrt{2}} \sum_{k=1}^\infty A_k(n)\sqrt{k} \cdot
\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)</math>
where
<math display="block">A_k(n) = \sum_{0 \le m < k, \; (m, k) = 1}
e^{ \pi i \left( s(m, k) - 2 nm/k \right) }.</math>
The [[multiplicative inverse]] of its generating function is the [[Euler function]]; by Euler's [[pentagonal number theorem]] this function is an alternating sum of [[pentagonal number]] powers of its argument.
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