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A generalization of Menon's identity with Dirichlet characters | | Yan Li
; Xiaoyu Hu
; Daeyeoul Kim
; | | Date: |
2 Feb 2018 | | Abstract: | The classical Menon’s identity [7] states that
egin{equation*}label{oldbegin1} sum_{substack{ainBbb 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 $Bbb Z_n=Bbb Z/nBbb
Z$, $gcd( , )$ 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.
In this paper, we generalize Menon’s identity with Dirichlet characters in
the following way: egin{equation*}
sum_{substack{ainBbb Z_n^ast b_1, ..., b_kinBbb Z_n}}
gcd(a-1,b_1, ..., b_k, n)chi(a)=varphi(n)sigma_kleft(frac{n}{d}
ight),
end{equation*} where $k$ is a non-negative integer and $chi$ is a Dirichlet
character modulo $n$ whose conductor is $d$.
Our result can be viewed as an extension of Zhao and Cao’s result [16] to
$k>0$.
It can also be viewed as an extension of Sury’s result [12] to Dirichlet
characters. | | Source: | arXiv, 1802.0531 | | Services: | Forum | Review | PDF | Favorites | |
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