On Erdős's Method for Bounding the Partition Function. Issue 8 (2nd September 2021)

• Record Type: Journal Article
• Authors: Antonir, Asaf Cohen; Shapira, Asaf
• Is Part Of: American Mathematical Monthly. Volume 128:Issue 8(2021)
• Page Start: 744
• Page End: 747
• Publication Date: 2021-09-02
• Subjects: MSC: Primary 11P81 -- Secondary 11P83 -- 05A17; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2021.1944756
• Languages: English
• ISSNs: 0002-9890

Abstract

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.