Dyson's ranks and Appell-Lerch sums
In [in: Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 7--28 (1988; Zbl 0652.10009)], \textit{F. J. Dyson} defined the rank of an integer partition to be its largest part minus its number of parts and conjectured that the partitions of \(5n + 4\) (resp. \(7n + 5\)) form \(5\) (resp. \(7\)) groups of equal size when sorted by their ranks modulo \(5\) (resp. \(7\)). This conjecture was proven ten years later by Atkin and Swinnerton-Dyer. In [Ann. Math. (2) 171, No. 1, 419--449 (2010; Zbl 1277.11096)], \textit{K. Bringmann} and \textit{K. Ono} used the theory of harmonic weak Maass form and showed that Dyson's rank partition function also satisfies congruences of Ramanujan type. In this paper, the authors invoked Appell-Lerch sums to prove a general formula for Dyson's ranks by considering the deviation of the ranks from the average \[ D(a,M)=\sum_{n=0}^{\infty} \left( N(a,M;n)-\frac{p(n)}{M}\right)q^n, \] where \(p(n)\) denotes the number of partitions of \(n\) and \(N(a,M;n)\) is the number of partitions of \(n\) with rank congruent to \(a \bmod M\).
scientific article