An Elementary Proof of the Transformation Formula for the Dedekind Eta Function

Abstract

In this work, we give an elementary proof of the transformation formula for the Dedekind eta function under the action of the modular group $\text{PSL}\,(2,\mathbb{Z})$. We start by giving a proof of the transformation formula $\eta(\tau)$ under the transformation $\tau\to -1/\tau$, using the Jacobi triple product identity and the Poisson summation formula. After we establish some identities for the Dedekind sum, the transformation formula for $\eta(\tau)$ under the transformation induced by a general element of the modular group $\text{PSL}\,(2,\mathbb{Z})$ is derived by induction.