A simplified proof of the partition formula
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 \(\varphi(z)=[(1-z)/2\pi]^{\tfrac 12} \exp| (\frac 1{12}\pi^2)(1+z)/(1-z)|\).
scientific article; zbMATH DE number 3172031