Armenian Journal of Mathematics, 16(4)
2024; 1–22
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An Elementary Proof of the Transformation Formula for the Dedekind Eta Function

Ze-Yong Kong*, Lee Peng Teo

Received: 2024-12-03 · Published: 2024-04-11

© 2024 The Author(s) and the original publisher.

CC BY 4.0 This work is licensed under a Creative Commons Attribution 4.0 International License.
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Original title
An Elementary Proof of the Transformation Formula for the Dedekind Eta Function
Authors
Ze-Yong Kong, Lee Peng Teo
Source journal
Armenian Journal of Mathematics, 16(4)
Published
2024-04-11
Licence
Creative Commons Attribution 4.0 International
Original
https://doi.org/10.52737/18291163-2024.16.4-1-22

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.
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