Congruences of Multipartition Functions Modulo Powers of Primes

TL;DR

This paper establishes new congruences for multipartition functions modulo prime powers by linking their generating functions to modular forms, extending classical results like Ramanujan's congruences.

Contribution

It introduces a novel modular form framework for multipartition functions, deriving new congruences and generalizing known results such as Ramanujan's and Gandhi's congruences.

Findings

Generating functions are congruent to modular form spaces modulo prime powers.

Reproduces classical Ramanujan congruences for partition functions.

Establishes new $m^k$-adic congruences using Hecke operators.

Abstract

Let $p_r(n)$ denote the number of $r$-component multipartitions of $n$, and let $S_{\gamma,\lambda}$ be the space spanned by $\eta(24z)^\gamma \phi(24z)$, where $\eta(z)$ is the Dedekind's eta function and $\phi(z)$ is a holomorphic modular form in $M_\lambda({\rm SL}_2(\mathbb{Z}))$. In this paper, we show that the generating function of $p_r(\frac{m^k n +r}{24})$ with respect to $n$ is congruent to a function in the space $S_{\gamma,\lambda}$ modulo $m^k$. As special cases, this relation leads to many well known congruences including the Ramanujan congruences of $p(n)$ modulo $5,7,11$ and Gandhi's congruences of $p_2(n)$ modulo 5 and $p_{8}(n)$ modulo 11. Furthermore, using the invariance property of $S_{\gamma,\lambda}$ under the Hecke operator $T_{\ell^2}$, we obtain two classes of congruences pertaining to the $m^k$-adic property of $p_r(n)$.