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Another generalization of Menon’s identity in the ring of algebraic integers

Volume 160 / 2020

Yujie Wang, Chungang Ji Colloquium Mathematicum 160 (2020), 213-221 MSC: Primary 11R04; Secondary 11A25. DOI: 10.4064/cm7785-6-2019 Published online: 27 January 2020

Abstract

Let $\varphi (n)$ be Euler’s totient function and $\tau (n)$ the divisor function. Recently, Zhao and Cao proved that $$ \sum _{\substack {a=1\atop \gcd (a, n)=1}}^{n} \gcd (a-1, n)\chi (a)=\varphi (n)\tau (n/d), $$ where $\chi $ is a Dirichlet character modulo $n$ with conductor $d$. We generalize the above identity to the ring of algebraic integers by considering arithmetical functions and characters.

Authors

  • Yujie WangSchool of Mathematics and Statistics
    Anhui Normal University
    Wuhu 241003
    People’s Republic of China
    e-mail
  • Chungang JiSchool of Mathematical Sciences
    Nanjing Normal University
    Nanjing 210023 People’s Republic of China
    e-mail

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