The Fourier expansion of \eta(z)\eta(2z)\eta(3z)/\eta(6z)
The Fourier expansion of \eta(z)\eta(2z)\eta(3z)/\eta(6z)
We compute the Fourier coefficients of the weight one modular form $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$ in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form $\eta(z)^4/\eta(2z)^2$. In the last section we use our main result to give an elementary proof of an identity by Victor Kac.
Comment: 10 pages. Version 2: Section 4 was added
- French National Centre for Scientific Research France
- Institut de Recherche Mathématique Avancée France
- University of Quebec Canada
- University of Strasbourg France
- University of Montreal Canada
Fourier coefficients of automorphic forms, Finite ground fields in algebraic geometry, Dedekind eta function, Mathematics - Number Theory, Enumerative problems (combinatorial problems) in algebraic geometry, Dedekind eta function, Dedekind sums, Fourier coefficient, 11F11, 11F20, 14C05, 14G15, 14N10, eta products, punctual Hilbert scheme, Parametrization (Chow and Hilbert schemes), [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Fourier coefficients of automorphic forms, Finite ground fields in algebraic geometry, Dedekind eta function, Mathematics - Number Theory, Enumerative problems (combinatorial problems) in algebraic geometry, Dedekind eta function, Dedekind sums, Fourier coefficient, 11F11, 11F20, 14C05, 14G15, 14N10, eta products, punctual Hilbert scheme, Parametrization (Chow and Hilbert schemes), [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
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