<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well.</p>
</abstract>

Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well.

Proofs, generalizations and analogs of Menon’s identity: a survey

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Dedicated to my father, László Gy. Tóth, high school mathematics teacher, on the occasion of his 90th birthday

Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n)...

Proofs, generalizations and analogs of Menon’s identity: a survey

Published Online: Nov 15, 2023

Page range: 142 - 197

Received: Jan 17, 2023

DOI: https://doi.org/10.2478/ausm-2023-0009

Keywords
Menon’s identity, arithmetic function, Euler’s function, Ramanujan’s sum, Dirichlet character, Dirichlet convolution, unitary convolution, group action, orbit counting lemma, n-even function, finite Fourier representation

© 2023 László Tóth, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Proofs, generalizations and analogs of Menon’s identity: a survey

Published Online: Nov 15, 2023

Page range: 142 - 197

Received: Jan 17, 2023

DOI: https://doi.org/10.2478/ausm-2023-0009

KeywordsMenon’s identity, arithmetic function, Euler’s function, Ramanujan’s sum, Dirichlet character, Dirichlet convolution, unitary convolution, group action, orbit counting lemma, n-even function, finite Fourier representation

© 2023 László Tóth, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Menon’s identity, arithmetic function, Euler’s function, Ramanujan’s sum, Dirichlet character, Dirichlet convolution, unitary convolution, group action, orbit counting lemma, n-even function, finite Fourier representation