2
$\begingroup$

In this Numberphile video, the partition function is discussed. At some point in the video, an approximation function is briefly talked about: enter image description here It is mentioned that the approximation equation has additional terms that can make it more accurate and in using four terms the approximation it will result in an answer accurate to the nearest integer for n>200. The video didn't link anything to this extended equation and I looked online briefly but couldn't find anything for this equation. Does anyone know where I could find it?

  • 0
    Only $4$ terms sufficient for every $n>200$ ? This would be awesome! Curious : For the fibonacci-numbers, we only need one term ... – 2017-01-17
  • 0
    Very unlikely a fixed number of extra terms give the nearest integer regardless of $n.$ Suggest Hardy and Wright book. – 2017-01-17

1 Answers 1

1

We find in section VIII.22 of Analytic Combinatorics by P. Flajolet and R. Sedgewick:

The number of integer partitions $p(n)$ satisfies the Hardy-Ramanujan-Rademacher expansion which is the exact formula \begin{align*} p(n)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^\infty A_k(n)\sqrt{k}\frac{d}{dn}\frac{\sinh \left(\frac{\pi}{k} \sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right)}{\sqrt{n-\frac{1}{24}}} \end{align*} where \begin{align*} A_k(n)=\sum_{{h \mod k}\atop{gcd(h,k)=1}}\omega_{h,k}e^{-2 i\pi h/k} \end{align*} and $\omega_{h,k}$ is the $24$th root of unity, $\omega_{h,k}=\exp(\pi i s(h,k))$ and \begin{align*} s_{h,k}=\sum_{\mu=1}^{k-1}\left\{\left\{\frac{\mu}{k}\right\}\right\} \left\{\left\{\frac{h\mu}{k}\right\}\right\} \end{align*} is known as a Dedekind sum, with $\{\{x\}\}=x-\lfloor x\rfloor -\frac{1}{2}$.

Some information is also stated in Wiki.