On the parity of partition functions
type: Item
instance of: scholarly article
title: On the parity of partition functions
zbMATH Open document ID: 0844.11065
published in: Illinois Journal of Mathematics
publication date: 1996-09-01T00:00:00Z
review text: Let \(p(n)\) denote the number of unrestricted partitions of \(n\), and \(q(n)\) the number of partitions of \(n\) into distinct parts. The main result of this paper is that the number of \(n\leq N\) such that \(p(n)\) is odd and the number of \(n\leq N\) such that \(p(n)\) is even are both \(>(\log N)^c\) for any \(c< 0.58\) and \(N\) large enough. This improves a result of \textit{L. Mirsky} [J. Math. Anal. Appl. 93, 593-598 (1983; Zbl 0513.10046)]\ where \((\log N)^c\) was replaced by \((\log \log N)/ 2\log 2\). Also it is proved that both \(p(n) \equiv q(n)\bmod 2\), \(n\leq N\) and \(p(n) \equiv q(n) +1 \bmod 2\), \(n\leq N\) have more than \((\log N)^c\) solutions for \(N\) large enough, and \(c< 1/2\). The proofs use classical identities in partition theory.
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 878715
zbMATH Keywords: parity of partition functions
zbMATH Keywords: number of unrestricted partitions
zbMATH Keywords: number of partitions
author: Jean-Louis Nicolas
author: András Sárközy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 878715