Simple upper bounds for partition functions
The author proves the following Theorem: Let \(p_k(n)\) denote the number of partitions of \(n\) with exactly \(k\) parts. There exists an absolute constant \(A<1\) such that for \(k<n\) \[ p_k(n)<A\frac{e^{c\sqrt{n-k}}}{(n-k)^{3/4}}e^{-\frac{2\sqrt{n-k}}{c}\text{Li}_2(e^{-\frac{c(k+1/2)}{2\sqrt{n-k}}})}, \] where \(\text{Li}_2(x)=\sum_{m=1}^\infty\frac{x^m}{m^2}\), \(|x|\leq 1\), is the dilogarithm function, and \(c=\pi\sqrt{2/3}\).
scientific article; zbMATH DE number 5636932
scientific article