The theta splitting function

Abstract

In this paper we study the Theta splitting function $\Theta (s+1)$, a function defined on the positive integers. We study the distribution of this function for sufficiently large values of the integers. As an application we show that \begin{align}\sum \limits_{m=0}^{s}\prod \limits_{\substack{0\leq j \leq m\\σ:[0,m]\rightarrow [0,m]\\σ(j)\neq σ(i)}}(s-σ(j))\sim s^s\sqrt{s}e^{-s}\sum \limits_{m=1}^{\infty}\frac{e^m}{m^{m+\frac{1}{2}}}.\nonumber \end{align} and that \begin{align}\sum \limits_{j=0}^{s-1}e^{-γj}\prod \limits_{m=1}^{\infty}\bigg(1+\frac{s-j}{m}\bigg)e^{\frac{-(s-j)}{m}}\sim \frac{e^{-γs}}{\sqrt{2π}}\sum \limits_{m=1}^{\infty}\frac{e^m}{m^{m+\frac{1}{2}}}.\nonumber \end{align}

4 pages