论文标题
关于对涉及Dirichlet角色的数字字段的概括
On a generalization of Menon-Sury identity to number fields involving a Dirichlet Character
论文作者
论文摘要
对于每个正整数$ n $，Sita Ramaiah的身份都指出\ Medskip \ begin {equation*} \ sum_ {a_1，a_1+a_1+a_2 \ in（\ mathbb {z}/n \ mathbb {z}） ϕ_2（n）σ_0（n）\; \ text {where} \; ϕ_2（n）= \ sum_ {a_1，a_2，a_1+a_2 \ in（\ mathbb {z}/n \ mathbb {z}）^*} 1，\ end end {equation*} \ medskip ring $ \ mathbb {z}/n \ mathbb {z} $和$σ_s（n）= \ displayStyle \ sum_ {d \ mid n} d^s $。 \ smallskip 这种身份也可以看作是梅农身份的概括。在本文中，我们将这种身份推广到涉及Dirichlet字符$χ$的代数数字字段$ k $。我们的结果是在\ cite {wj}和\ cite {sury}中进一步概括了最新结果。
For every positive integer $n$, Sita Ramaiah's identity states that \medskip \begin{equation*} \sum_{a_1, a_2, a_1+a_2 \in (\mathbb{Z}/n\mathbb{Z})^*} \gcd(a_1+a_2-1,n) = ϕ_2(n)σ_0(n) \; \text{ where } \; ϕ_2(n)= \sum_{a_1, a_2, a_1+a_2 \in (\mathbb{Z}/n\mathbb{Z})^*} 1, \end{equation*} \medskip where $(\mathbb{Z}/n\mathbb{Z})^*$ is the multiplicative group of units of the ring $\mathbb{Z}/n\mathbb{Z}$ and $σ_s(n) = \displaystyle\sum_{d\mid n}d^s$. \smallskip This identity can also be viewed as a generalization of Menon's identity. In this article, we generalize this identity to an algebraic number field $K$ involving a Dirichlet character $χ$. Our result is a further generalization of a recent result in \cite{wj} and \cite{sury}.
