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$\\gcd(\\ ,\\ )$ represents the greatest common divisor, $\\varphi(n)$ is the Euler's totient function and $\\sigma_{k} (n) =\\sum_{d|n } d^{k}$ is the divisor function.\n  In this paper, we generalize Menon's identity with Dirichlet characters in the following way: \\begin{equation*}\n  \\sum_{\\substack{a\\in\\Bbb 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generalization of Menon's identity with Dirichlet characters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daeyeoul Kim, Xiaoyu Hu, Yan Li","submitted_at":"2018-02-02T01:43:57Z","abstract_excerpt":"The classical Menon's identity [7] states that\n  \\begin{equation*}\\label{oldbegin1} \\sum_{\\substack{a\\in\\Bbb Z_n^\\ast }}\\gcd(a -1,n)=\\varphi(n) \\sigma_{0} (n), \\end{equation*} where for a positive integer $n$, $\\Bbb Z_n^\\ast$ is the group of units of the ring $\\Bbb Z_n=\\Bbb Z/n\\Bbb Z$, $\\gcd(\\ ,\\ )$ represents the greatest common divisor, $\\varphi(n)$ is the Euler's totient function and $\\sigma_{k} (n) =\\sum_{d|n } d^{k}$ is the divisor function.\n  In this paper, we generalize Menon's identity with Dirichlet characters in the following way: \\begin{equation*}\n  \\sum_{\\substack{a\\in\\Bbb 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