# Ramanujan Congruences for Fractional Partition Functions

## TL;DR

This paper extends Ramanujan-type congruences to fractional partition functions, providing a general framework to find such congruences modulo any integer, and proves new specific congruences.

## Contribution

It introduces a unified approach using non-ordinary primes to characterize and discover congruences for fractional partition functions beyond classical cases.

## Key findings

- Established a general framework for congruences modulo any integer.
- Proved new congruences such as for $p_{57/61}(17^2n-3)$ modulo $17^2$.
- Extended Ramanujan's classical congruences to fractional partition functions.

## Abstract

For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form $p(\ell n + c)\equiv 0 \pmod{\ell}$ for a prime $\ell$ and integer $c$ were studied by Ramanujan. Such congruences exist only for $\ell\in\{5,7,11\}.$ Chan and Wang [4] recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as $p_\frac{57}{61}(17^2n-3)\equiv 0 \pmod{17^2}$.