About the Hardy-Ramanujan partition function asymptotics

TL;DR

This paper revisits the Hardy-Ramanujan partition function asymptotics, employing the Laplace transform approach to derive leading order results and extending the method with additional examples.

Contribution

It introduces a Laplace transform-based derivation of partition function asymptotics, providing a new perspective and extending the probabilistic approach.

Findings

Derived leading order asymptotics using Laplace transform

Extended the Laplace transform method with additional examples

Provided a new perspective on classical partition asymptotics

Abstract

The Hardy-Ramanujan partition function asymptotics is a famous result in the asymptotics of combinatorial sequences. It was originally derived using complex analysis and number-theoretic ideas by Hardy and Ramanujan. It was later re-derived by Paul Erd\H{o}s using real analytic methods. Later still, D.J.~Newman used just the usual Hayman saddle-point approach, ubiquitous in asymptotic analysis. Fristedt introduced a probabilistic approach, which was further extended by Dan Romik, for restricted partition functions. Our perspective is that the Laplace transform changes the essentially algebraic generating function into an exponential form. Using this, we carry out the exercise of deriving the leading order asymptotics, following the Fristedt-Romik approach. We also give additional examples of the Laplace transform method.