The 26th power of Dedekind's η-function

Abstract

We prove a simple and explicit formula, which expresses the 26th power of Dedekind's η-function as a double series. The proof relies on properties of Ramanujan's Eisenstein series P, Q and R, and parameters from the theory of elliptic functions. The formula reveals a number of properties of the product ∏ j = 1 ∞ ( 1 − q j ) 26 , for example its lacunarity, the action of the Hecke operator, and sufficient conditions for a coefficient to be zero.