Where can I find a proof of the asymptotic expresion of partition numbers by Hardy-Ramanujan?

Question

I'm starting to study number theory and I´m interested in partitions, but I don't find a proof of this asymptotic expression $p(n)$ given by Hardy-Ramanujan.

Answer 1

Here are two references which might be useful.

• Modular Functions and Dirichlet Series in Number Theory by T. Apostol

  Chapter 5: Rademacher's series for the partition function is devoted to the theme culminating in:

  Theorem 5.10: If $n\geq 1$ the partion function $p(n)$ is represented by the convergent series

  \begin{align*} \color{blue}{p(n)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}A_k(n)\sqrt{k}\frac{d}{dn} \left(\frac{\sinh\left\{\frac{\pi}{k}\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right\}}{\sqrt{n-\frac{1}{24}}}\right)}\tag{1} \end{align*} where \begin{align*} \color{blue}{A_k(n)=\sum_{{0\leq h<k}\atop{(h,k)=1}}e^{\pi i s(h,k)-2\pi i nh/k}} \end{align*} and with $s(h,k)$ being the Dedekind sum \begin{align*} s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\left[\frac{hr}{k}\right]-\frac{1}{2}\right) \end{align*}

• The Theory of Partitions by G. E. Andrews

  Here is also chapter 5: The Hardy-Ramanujan-Rademacher Expansion of $p(n)$ devoted to the theme. The theorem (1) is presented with a detailed proof in roughly 20 pages. G. E. Andrews has also added very nice and interesting aspects about the development of this remarkable formula for $p(n)$. He ends the historical notes with:

  • (verbatim from the book): We owe the theorem to a singularly happy collaboration of two men, of quite unlike gifts, in which each contributed the best, most characteristic, and most fortunate work that was in him. Ramanujan's genius did have this one opportunity worthy of it.

Answer 2

A simple search for "hardy ramanujan partitions" comes up with many answers.

Some are:

Ramanujan's collected works. The paper you want is "Asymptotic formulæ in combinatory analysis".

https://www.amazon.com/Collected-Srinivasa-Ramanujan-Chelsea-Publication/dp/0821820761

A very nice discussion of the theorem and its history.

https://arxiv.org/pdf/2003.06908.pdf

Another good discussion of the history is in

https://www.tandfonline.com/doi/full/10.1080/00029890.2017.1389178

and its pdf form

https://www.tandfonline.com/doi/epdf/10.1080/00029890.2017.1389178?needAccess=true&role=button

There are others.

Have fun.