The 2-adic behavior of the number of partitions into distinct parts
Let \(Q(n)\) be the number of partitions of a nonnegative integer into \(n\) distinct parts. In a series of papers in 1996-97 K. Alladi examined the 2-adic behavior of \(Q(n)\) and in 1997 \textit{B. Gordon} and \textit{K. Ono} [Ramanujan J. 1, No. 1, 25-34 (1997; Zbl 0907.11036)] showed that if \(j\) is a positive integer then \(Q(n)\equiv 0\pmod{2^j}\) for a subset of the nonnegative integers \(n\) with arithmetic density one. Here the authors show, using modular forms, that if \(i\not\equiv 0\pmod{2^j}\), then \(\#\{0\leq n\leq X\mid Q(n)\equiv i\pmod{2^j}\}\gg_j\sqrt X/\log X\). A strong form of this can be proved for even \(i\). A rather more complicated statement is proved for \(Q(n)\pmod 8\), using the arithmetic of the ring \(\mathbb{Z}[\sqrt{-6}]\).
scientific article; zbMATH DE number 1569470