Deviation of the rank and crank modulo 11 (Q6596364)

In the present paper, the author establishes explicit \(11\)-dissection expressions for the deviations of the rank and crank modulo \(11\) in terms of three auxiliary functions introduced by the author. As applications, the author proves a number of rank--crank inequalities as well as congruences for rank and crank moments, Andrews' spt-function and Eisenstein series; some of these results were previously derived by \textit{A. O. L. Atkin} and \textit{P. Swinnerton-Dyer} [Proc. Lond. Math. Soc. (3) 4, 84--106 (1954; Zbl 0055.03805)], \textit{A. O. L. Atkin} and \textit{S. M. Hussain} [Trans. Am. Math. Soc. 89, 184--200 (1958; Zbl 0083.26201)], \textit{F. G. Garvan} [Trans. Am. Math. Soc. 305, No. 1, 47--77 (1988; Zbl 0641.10009)], \textit{A. B. Ekin} [J. Comb. Theory, Ser. A 83, No. 2, 283--289 (1998; Zbl 0909.05010)], \textit{A. Berkovich} and \textit{F. G. Garvan} [J. Number Theory 128, No. 6, 1731--1748 (2008; Zbl 1143.05008)], and \textit{S. Chern} [Int. J. Number Theory 18, No. 1, 141--163 (2022; Zbl 1491.11095)], while the author offers alternative proofs.