A partition of an integer $n$ is a way of writing $n$ as a sum of integers. The partition function $p(n)$ counts the number of distinct partitions of $n$.
In 1918, Hardy and Ramanujan proved the amazing result
$$p(n) \sim \frac {1} {4n\sqrt3} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right)$$
This is one of those formulae that to the uninitiated look like witchcraft. I feel a burning desire to understand why it is true, but number theory is not my forte and I don't really fancy digging up a bunch of papers a century old, even a single one of which can often take days to find.
Is there a modern exposition that cuts to this result with as little groundwork as possible, or, even better, an argument that fits into a MSE answer that can at least outline how that asymptotic expression is obtained?
Best Answer
The proof of the asymptotic formula for the partition function given by Hardy and Ramanujan was "the birth of the circle method", and used properties of modular forms. Erdös was trying to give a proof with elementary methods (he also gave a so-called elementary proof of the PNT with Selberg). He succeeded in 1942 to give such a proof, but only with "unknown" constant $a$, see here. Afterwards Newman gave a "simplified proof", and obtained also that $a=\frac{1}{4n\sqrt{3}}$, see here.
There are several "modern" references now, which give an elementary proof of the asymptotic formula for $p(n)$. Here are two references:
M. B. Nathanson: On Erdös's elementary method in the asymptotic theory of partitions.
Daniel M. Kane (was misspelled as "Cane"): An elementary derivation of the asymptotic of the partition function.
Instead of using modular forms etc. the elementary method of Erdös only uses elementary estimates of exponential sums and an induction from the identity $$ np(n)=\sum_{ka\le n}ap(n-ka). $$