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

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Submitted: 2024-12-02; Published: 2024-12-02

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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 PSL(2,Z)PSL(2,Z). We start by giving a proof of the transformation formula η(τ)η(τ) under the transformation τ→−1/ττ→−1/τ, using the Jacobi triple product identity and the Poisson summation formula. After we establish some identities for the Dedekind sum, the transformation formula for η(τ)η(τ) under the transformation induced by a general element of the modular group PSL(2,Z)PSL(2,Z) is derived by induction.