Ramanujan Congruences for Fractional Partition Functions

Abstract

For rational $\alpha$, the fractional partition functions $p_{\alpha }(n)$ are given by the coefficients of the generating function $(q;q{)}_{\infty }^{\alpha }$. When $\alpha =-1$, one obtains the usual partition function. Congruences of the form $p(\ell n+c)\equiv 0(\mathrm{mod}\ell )$ 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 $p_{\frac{57}{61}}(17^{2}n-3)\equiv 0(\mathrm{mod}17^2)$.

13 pages