Equidistribution and inequalities for partitions into powers

2020/02/13 by Alexandru Ciolan, Ciolan, Alexandru
Mathematics · #11P82 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2002.05682

openalex publication_date 2020/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If p_k(a,m,n) denotes the number of partitions of n into kth powers with a number of parts that is congruent to a modulo m, then p₂(0,2,n)∼ p₂(1,2,n) and the sign of the difference p₂(0,2,n)- p_k(1,2,n) alternates with the parity of n, as proven by recent work of the author (2020). In this paper, we place the problem in a broader framework. By analytic arguments using the circle method and Gauss sums estimates, we show that the same results hold for any k≥2. By combinatorial arguments, we show that the sign of the difference p_k(0,2,n)- p_k(1,2,n) depends on the parity of n for a larger class of partitions.

Citations

• New upper bounds on sphere packings I

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