Menon's identity: proofs, generalizations and analogs

Menon’s identity states that for every positive integer $n$ one has $\sum (a − 1, n) = \phi(n)\tau(n)$, where $a$ runs through a reduced residue system $({\rm mod}\ n)$,$(a−1, n)$ stands for the greatest common divisor of $a − 1$ and $n$, $\phi(n)$ is Euler’s totient function and $\tau(n)$ is the number of divisors of $n$. It is named after Puliyakot Kesava Menon, who proved it in 1965. Menon’s identity has been the subject of many research papers, also in the last years.

In this talk I will present different methods to prove this identity, and will point out those that I could not identify in the literature. Then I will survey the directions to obtain generalizations and analogs. I will also present some of my own general identities.

Laszlo Toth