Lacunarity of Dedekind η-products

The Dedekind η-function is defined by where τ lies in the upper half plane ℋ = {tau;|Im(τ) > 0}, and x = e 2πiτ . It is a modular form of weight ½ with a multiplier system. We define an η-product to be a function f (τ) of the form where r δ ε ℤ. This is a modular form of weight with a multiplier system. The Fourier coefficients of η-products are related to many well-known number-theoretic functions, including partition functions and quadratic form representation numbers. They also arise from representations of the “monster” group [3] and the Mathieu group M 24 [13]. The multiplicative structure of these Fourier coefficients has been extensively studied. Recent papers include [1], [4], [5] and [6]. Here we study the connections between the density of the non-zero Fourier coefficients of f (τ) and the representability of f (τ) as a linear combination of Hecke character forms (defined in Section 4 below). We first make the following definition.