Moments of ranks and cranks, and Quotients of Eisenstein Series and the Dedekind Eta Function

Abstract

Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote the $k$-th moments of ranks and cranks in the theory of partitions. Let $e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(\tau )/\eta (\tau )$, where $E_{2r}(\tau )$ is the classical Eisenstein series of weight $2r$ and $\eta (\tau )$ is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for $k\le 5$, $N_k(n)$ and $M_k(n)$ can be expressed using $e_{2r}(n)$ ($2\le r\le k$), $p(n)$ and $N_2(n)$. For $k>5$, additional functions are required for such expressions. For $r\in \{2,3,4,5,7\}$, by studying the action of Hecke operators on $E_{2r}(\tau )/\eta (\tau )$, we provide explicit congruences modulo arbitrary powers of primes for $e_{2r}(n)$. Moreover, for $\ell \in \{5,7,11,13\}$ and any $k\ge 1$, we present uniform methods for finding nice representations for ${\sum }_{n=0}^{\infty }{e}_{2r}\left(\frac{{\ell }^{k}n+1}{24}\right)q^n$, which work for every $r\ge 2$. These representations allow us to prove congruences modulo powers of $\ell$, and we have done so for $e_4(n)$ and $e_6(n)$ as examples. Based on the congruences satisfied by $e_{2r}(n)$, we establish congruences modulo arbitrary powers of $\ell$ for the moments and symmetrized moments of ranks and cranks as well as higher order $\mathrm{spt}$-functions.

arXiv comment52 pages