On Ramanujan congruences for modular forms of integral and half-integral weights

In 1916 Ramanujan observed a remarkable congruence: . The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number . In this paper we give a simple proof of the Ramanujan congruence and its generalizations to forms of higher integral and half-integral weights.