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.
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