Rademacher's Formula for the Partition Function

TL;DR

This paper revisits Rademacher's formula for the partition function, providing detailed derivations and asymptotic analysis, along with numerical results to illustrate the convergence and accuracy of the series.

Contribution

It offers a comprehensive pedagogical derivation of Rademacher's formula and derives the leading asymptotic behavior of the partition function p(n).

Findings

Detailed derivation of Rademacher's series for p(n)

Asymptotic formula for p(n) as n approaches infinity

Numerical tables demonstrating convergence and accuracy

Abstract

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details given. We also derive the leading asymptotic behavior of $p(n)$ when $n$ approaches infinity. Some numerical results are tabled.