On the parity of additive representation functions
Let \(p(n)\) denote the number of partitions of \(n\). It is shown that \(p(n)\) is even for \(\gg\sqrt{N}\) values of \(n\leq N\), and is odd for \(\gg N^{\frac 12+o(1)}\) values of \(n\leq N\). These results improve those of \textit{J.-L. Nicolas} and \textit{A. Sárközy} [Ill. J. Math. 39, 586-597 (1995; Zbl 0844.11065)]. Further, let \(p({\mathcal A},n)\) denote the number of partitions of \(n\) with parts in \({\mathcal A}\). It is shown that there exists a set \({\mathcal A}\) of positive integers, with counting function \(A(x)\gg \frac{x}{\log x}\), such that \(p({\mathcal A},n)\) is even for all \(n\geq 4\). Another related result concerns \(r({\mathcal A},n)\), the number of solutions of \(a_i+ a_j=n\) with \(a_i,a_j\in {\mathcal A}\), \(i\leq j\), where \({\mathcal A}= \{a_1< a_2<\dots\}\). It is shown that there exists an infinite set \({\mathcal A}\subset \mathbb{N}\) with \(\limsup_{x\to\infty} \frac{A(x)} {(\log x)^2}< \infty\), and \(r({\mathcal A},n)\neq 1\) for all sufficiently large \(n\).
parity of additive representation functions
scientific article; zbMATH DE number 1247924