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

## TL;DR

This paper explores the relationships between moments of ranks and cranks in partition theory and Fourier coefficients of Eisenstein series over the Dedekind eta function, establishing explicit congruences and representations using quasi-modular forms and Hecke operators.

## Contribution

It provides new explicit formulas and congruences for moments of ranks and cranks via Eisenstein series and eta quotients, extending previous work for higher moments and primes.

## Key findings

- Expressed moments of ranks and cranks using Fourier coefficients of Eisenstein series for k ≤ 5.
- Derived explicit congruences modulo prime powers for these Fourier coefficients.
- Established congruences for moments and higher order spt-functions based on properties of these coefficients.

## 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 $k \leq 5$, $N_k(n)$ and $M_k(n)$ can be expressed using $e_{2r}(n)$ ($2\leq r \leq k$), $p(n)$ 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 $k\geq 1$, 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 $r\geq 2$. 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.