On the parity of partition functions
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.
scientific article; zbMATH DE number 878715