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). We start by giving a proof of the transformation formula η(τ) under the transformation τ→ -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) under the transformation induced by a general element of the modular group PSL\,(2,Z) is derived by induction.
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Kong et al. (2024) studied this question.
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