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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 () under the transformation -1/, using the Jacobi triple product identity and the Poisson summation formula. After we establish some identities for the Dedekind sum, the transformation formula for () under the transformation induced by a general element of the modular group PSL\, (2, Z) is derived by induction.
Kong et al. (Thu,) studied this question.