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Lacunarity of Dedekind η-products

Published online by Cambridge University Press:  18 May 2009

Basil Gordon  and
Sinai Robins
Basil Gordon
Affiliation:
University of California, Los Angeles
Sinai Robins
Affiliation:
University of Northern Colorado, Greeley
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The Dedekind η-function is defined by

where τ lies in the upper half plane ℋ = {tau;|Im(τ) > 0}, and x = e2πiτ. 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 rδ 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 M24 [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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 1 , January 1995 , pp. 1 - 14
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

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