On Erd\H{o}s's Method for Bounding the Partition Function

TL;DR

This paper presents a simpler and more concise proof of Nathanson's bound on the growth rate of the partition function for integers constrained by modular conditions, originally derived using Erdős's method.

Contribution

It provides a shorter, more straightforward proof of Nathanson's bound, improving understanding of Erdős's method for partition function bounds.

Findings

Simplified proof of Nathanson's bound

Clarification of Erdős's method application

Enhanced understanding of partition function bounds

Abstract

For fixed $m$ and $R\subseteq \{0,1,\ldots,m-1\}$, take $A$ to be the set of positive integers congruent modulo $m$ to one of the elements of $R$, and let $p_A(n)$ be the number of ways to write $n$ as a sum of elements of $A$. Nathanson proved that $\log p_A(n) \leq (1+o(1)) \pi \sqrt{2n|R|/3m}$ using a variant of a remarkably simple method devised by Erd\H{o}s in order to bound the partition function. In this short note we describe a simpler and shorter proof of Nathanson's bound.