Abstract
Menon’s identity is a classical identity involving gcd sums and the Euler totient function \(\phi \). In a recent paper, Zhao and Cao (Int. J. Number Theory 13(9) (2017) 2373–2379) derived the Menon-type identity \(\sum _{\begin{array}{c} k=1 \end{array}}^{n}(k-1,n)\chi (k) = \phi (n)\tau (\frac{n}{d})\), 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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Acknowledgements
The first author thanks the University Grants Commission of India for providing financial support for carrying out research work through their Junior Research Fellowship (JRF) scheme. The third author thanks the Kerala State Council for Science, Technology and Environment, Thiruvananthapuram, Kerala, India for providing financial support for carrying out research work.
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Chandran, A., Namboothiri, K.V. & Thomas, N.E. A Menon-type identity concerning Dirichlet characters and a generalization of the gcd function. Proc Math Sci 131, 18 (2021). https://doi.org/10.1007/s12044-021-00609-8
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DOI: https://doi.org/10.1007/s12044-021-00609-8