A simplified proof of the partition formula
type: Item
instance of: scholarly article
title: A simplified proof of the partition formula
zbMATH Open document ID: 0105.26701
published in: Michigan Mathematical Journal
publication date: 1962-01-01T00:00:00Z
review text: The author gives a simple and elegant proof of the Hardy-Ramanujan formula \[ p(n)\sim (1/4n\sqrt 3) \exp\pi\sqrt{2n/3}, \] where the partition function \(p(n)\) is defined by \[ f(z)=\sum_{n=0}^\infty p(n)z^n=\prod_{n=1}^\infty (1-z^n)^{-1}. \] The method is the customary circle method, using just one major and one minor arc. The proof is made to depend on the inequalities \[ | 1-z|\leq 2(1-| z|),\quad | f(z)|<\exp\{(1-| z|^{-1}+| 1-z|^{-1}\},\quad f(z)=\varphi(z)(1+O(1-z)), \] where \(\varphi(z)=[(1-z)/2\pi]^{\tfrac 12} \exp| (\frac 1{12}\pi^2)(1+z)/(1-z)|\).
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 3172031
zbMATH Keywords: proof of the Hardy-Ramanujan formula
zbMATH Keywords: partition function
zbMATH Keywords: circle method
author: Donald J. Newman
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1028998729
OpenAlex ID: W2084811867
DOI: 10.1307/MMJ/1028998729
description: scientific article; zbMATH DE number 3172031