On Erdős’s Method for Bounding the Partition Function

For fixed m and (Formula presented.), take A to be the set of positive integers congruent modulo m to one of the elements of R, and let (Formula presented.) be the number of ways to write n as a sum of elements of A. Nathanson proved that (Formula presented.) 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.