Ramanujan's partition generating functions modulo $\ell$
Kathrin Bringmann, William Craig, Ken Ono
TL;DR
This work addresses Ramanujan-type partition congruences modulo primes $\ell\ge5$ by deriving a closed-form expression for $P_\ell(q)$ in $\mathbb{F}_\ell[[q]]$, namely $P_\ell(q)\equiv c_\ell \frac{T_\ell(q)}{(q^\ell;q^\ell)_\infty} \pmod{\ell}$, where $T_\ell(q)$ records Hecke traces of weight $\ell-1$ cusp forms. The approach combines generalized pentagonal recurrences for $p(n)$ (via modular forms $R_k(z)$ built from $\eta$) with a detailed congruence analysis of the Hecke-trace generating function $T_{\ell}(q)$ and a matching theta-series to align generating functions modulo $\ell$. Key contributions include a new proof of Ramanujan's congruences for $\ell=5,7,11$ (where no nontrivial cusp forms exist in the relevant weights) and a uniform, explicit expression for $P_\ell(q)$ in terms of $\ell$-ramified Hecke data, with the constant $c_\ell$ given modulo $\ell$. This work deepens the link between partition arithmetic and the theory of modular forms, offering a practical framework to derive partition congruences modulo primes using Hecke-theoretic inputs.
Abstract
For the partition function $p(n)$, Ramanujan proved the striking identities $$ P_5(q):=\sum_{n\geq 0} p(5n+4)q^n =5\prod_{n\geq 1} \frac{\left(q^5;q^5\right)_{\infty}^5}{(q;q)_{\infty}^6}, $$ $$ P_7(q):=\sum_{n\geq 0} p(7n+5)q^n =7\prod_{n\geq 1}\frac{\left(q^7;q^7\right)_{\infty}^3}{(q;q)_{\infty}^4}+49q \prod_{n\geq 1}\frac{\left(q^7;q^7\right)_{\infty}^7}{(q;q)_{\infty}^8}, $$ where $(q;q)_{\infty}:=\prod_{n\geq 1}(1-q^n).$ As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes $\ell \geq 5,$ closed form expressions of the power series $$ P_{\ell}(q):=\sum_{n\geq 0} p(\ell n-δ_{\ell})q^n\pmod{\ell}, $$ where $δ_{\ell}:=\frac{\ell^2-1}{24}.$ In this paper, we prove that $$ P_{\ell}(q)\equiv c_{\ell} \frac{T_{\ell}(q)}{ (q^\ell; q^\ell )_\infty} \pmod{\ell}, $$ where $c_{\ell}\in \mathbb{Z}$ is explicit and $T_{\ell}(q)$ is the generating function for the Hecke traces of $\ell$-ramified values of special Dirichlet series for weight $\ell-1$ cusp forms on $SL_2(\mathbb{Z})$. This is a new proof of Ramanujan's congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10.