Moments of ranks and cranks, and quotients of Eisenstein series and the Dedekind eta function

Abstract

Atkin and Garvan introduced the functions Nk(n) and Mk(n), which denote the k-th moments of ranks and cranks in the theory of partitions. Let e2r(n) be the n-th Fourier coefficient of E2r(τ)/η(τ), where E2r(τ) is the classical Eisenstein series of weight 2r and η(τ) is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for k≤5, Nk(n) and Mk(n) can be expressed using e2r(n) (0≤r≤k), p(n) and N2(n). For k>5, additional functions are required for such expressions. For r∈{2,3,4,5,7}, by studying the action of Hecke operators on E2r(τ)/η(τ), we provide explicit congruences modulo arbitrary powers of primes for e2r(n). Moreover, for ℓ∈{5,7,11,13} and any k≥1, we present uniform methods for finding nice representations for ∑n=0∞e2r(ℓkn+124)qn, which work for every r≥2. These representations allow us to prove congruences modulo powers of ℓ, and we have done so for e4(n) and e6(n) as examples. Based on the congruences satisfied by e2r(n), we establish congruences modulo arbitrary powers of ℓ for the moments and symmetrized moments of ranks and cranks as well as higher order spt-functions.