The Fourier expansion of \eta(z)\eta(2z)\eta(3z)/\eta(6z)

TL;DR

This paper computes the Fourier coefficients of a specific weight one modular form involving eta functions, relating them to representations as sums of two squares and connecting to known modular forms and identities.

Contribution

It provides an explicit Fourier expansion of a particular eta quotient modular form and links it to classical number theory and modular identities.

Findings

Fourier coefficients expressed via sums of two squares

Established relation to eta(z)^4/eta(2z)^2

Elementary proof of Kac's identity

Abstract

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.