About the Hardy-Ramanujan partition function 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.