Non-Existence of Ramanujan Congruences in Modular Forms of Level Four

Abstract

AbstractRamanujan famously found congruences like p(5n+4) ≡ 0 mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Г1(4) that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.