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arXiv · 2005.12509

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

Arya Chandran·Neha Elizabeth Thomas·K Vishnu Namboothiri
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Abstract

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.

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Arya Chandran, Neha Elizabeth Thomas, K Vishnu Namboothiri. 2020-05-26. A Menon-type Identity concerning Dirichlet characters and a generalization of the gcd function. https://arxiv.org/abs/2005.12509

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