{"paper":{"title":"A 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 Z_n^\\ast"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00531","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}