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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?

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    I think, that Erdös method is quite elementary - see here, and the references therein. – Dietrich Burde Oct 28 '14 at 16:07
  • @DietrichBurde That is a great reference. I'm a little confused though: did Erdős prove an asymptotic for $p(n)$ (as seemingly claimed in Nathanson), an asymptotic-with-unknown-constant for $p(n)$ (as it claims in Erdős here), or an asymptotic for $\log p(n)$ (in the style of the main theorem of Nathanson)? – Erick Wong Oct 29 '14 at 03:13
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    @ErickWong Erdös first proved the estimate for $p(n)$ with unknown constant in $1942$, then his method was refined by Don Newman - "A Simplified Proof of the Partition Formula"-which gave also the explicit constant. – Dietrich Burde Oct 29 '14 at 19:19
  • Recently a question which seems related have been posted on MathOverflow. – Martin Sleziak Jan 11 '17 at 12:10

1 Answers1

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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 $$ np(n)=\sum_{ka\le n}ap(n-ka). $$

Dietrich Burde
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  • The Newman proof is exactly what I was looking for. The generating function approach is quite elegant too. Amazingly, it only references one result from another publication! –  Oct 30 '14 at 07:29
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    Newman actually uses complex analysis; he's not refining Erdös's proof, but rather simplifying the original approach of Hardy and Ramanujan. You can take a look at Newman's paper and see for yourself. – Yuval Filmus May 20 '16 at 22:44
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    can someone point out why the use of modular forms could be dropped? – Student Jul 18 '19 at 13:16