On Erdős's Method for Bounding the Partition Function. Issue 8 (2nd September 2021) - Antonir, Asaf Cohen
Abstract: For fixed m and R ⊆ { 0, 1, …, 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 ) ≤ ( 1 + o ( 1 ) ) π 2 n | R | / 3 m using a variant of a remarkably simple method devised by Erdős in order to bound the partition function. In this short note, we describe a simpler and shorter proof of Nathanson's bound.