A Menon-type Identity concerning Dirichlet characters and a generalization of the gcd function

26 May 2020  ·  Arya Chandran, Neha Elizabeth Thomas, K Vishnu Namboothiri ·

Menon's identity is a classical identity involving gcd sums and the Euler totient function $\phi$. In a recent paper, Zhao and Cao derived the Menon-type identity $\sum\limits_{\substack{k=1}}^{n}(k-1,n)\chi(k) = \phi(n)\tau(\frac{n}{d})$, where $\chi$ is a Dirichlet character mod $n$ with conductor $d$... We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao-Cao identity can be improved if one uses the method we employ here. read more

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Number Theory 11A07, 11A25