On elementary estimates for the partition function

TL;DR

This paper derives bounds for the partition function using simple geometric inequalities and extends these methods to related functions, providing new elementary estimates in number theory.

Contribution

Introduces elementary geometric methods to estimate the partition function and its generalizations, offering a novel approach compared to traditional techniques.

Findings

Established upper and lower bounds for p(n).

Extended the method to generalized partition functions.

Demonstrated the effectiveness of geometric inequalities in number theory.

Abstract

In this paper, we obtain upper and lower bounds for the partition function $p(n)$ by using an elementary geometric inequality in Euclidean space, and we extend the method to generalizations of the partition function.