Will the real Hardy-Ramanujan formula please stand up?

TL;DR

This paper clarifies the original Hardy-Ramanujan formula for partition numbers, distinguishing it from simplified or later versions, and provides a detailed justification for these differences.

Contribution

It offers a clear, detailed analysis that differentiates the original Hardy-Ramanujan formula from its simplified and alternative forms, clarifying historical and mathematical distinctions.

Findings

Original Hardy-Ramanujan formula is distinct from simplified versions.

Provides a detailed justification for the differences among formulas.

Clarifies common misconceptions about the formulas' origins.

Abstract

The Hardy-Ramanujan formula for the number of integer partitions of $n$ is one of the most popular results in partition theory. While the unabridged final formula has been celebrated as reflecting the genius of its authors, it has become all too common to attribute either some simplified version of the formula which is not as ingenious, or an alternative more elegant version which was expanded on afterwards by other authors. We attempt to provide a clear and compelling justification for distinguishing between the various formulas and simplifications, with a summarizing list of key take-aways in the final section.