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

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An Elementary Proof of the Transformation Formula for the Dedekind Eta Function Ze-Yong Kong

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.

DOI: 10.52737/18291163-2024.16.4-1-22 Armenian Journal of Mathematics, 16(4) 1-22

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Keywords

Dedekind Eta Function, Transformation Formula, Modular Group, Functional Equation

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