On Ramanujan Congruences for Modular Forms of Integral and Half-integral Weights

In 1916 Ramanujan observed a remarkable congruence: τ(n) ≡ σ11(n) mod 691. 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 12 and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number B12. In this paper we give a simple proof of the Ramanujan congruence and its generalizations to forms of higher integral and half-integral weights. 0. Introduction Let τ(n) be given by q ∏∞ n=1(1 − q) = ∑∞ n=1 τ(n)q , and let σ11(n) = ∑ d|n d . Ramanujan [10] observed that τ(n) ≡ σ11(n) mod 691. The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form