# The theta splitting function

## TL;DR

This paper investigates the properties and distribution of the Theta splitting function, providing asymptotic formulas and applications related to sums involving this function for large integers.

## Contribution

It introduces and analyzes the Theta splitting function, deriving new asymptotic expressions and demonstrating their applications in summation formulas for large integers.

## Key findings

- Derived asymptotic formulas for the Theta splitting function.
- Established connections between the function and exponential sum approximations.
- Provided applications demonstrating the function's distribution properties.

## 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\\\sigma:[0,m]\rightarrow [0,m]\\\sigma(j)\neq \sigma(i)}}(s-\sigma(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^{-\gamma j}\prod \limits_{m=1}^{\infty}\bigg(1+\frac{s-j}{m}\bigg)e^{\frac{-(s-j)}{m}}\sim \frac{e^{-\gamma s}}{\sqrt{2\pi}}\sum \limits_{m=1}^{\infty}\frac{e^m}{m^{m+\frac{1}{2}}}.\nonumber \end{align}