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
Dedekind eta function, Mathematics - Number Theory, Dedekind eta function, Dedekind sums, 11F11, 11F20, 14C05, 14G15, 14N10, eta products, Parametrization (Chow and Hilbert schemes), Fourier coefficients of automorphic forms, Finite ground fields in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Fourier coefficient, punctual Hilbert scheme, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Dedekind eta function, Mathematics - Number Theory, Dedekind eta function, Dedekind sums, 11F11, 11F20, 14C05, 14G15, 14N10, eta products, Parametrization (Chow and Hilbert schemes), Fourier coefficients of automorphic forms, Finite ground fields in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Fourier coefficient, punctual Hilbert scheme, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
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