On a generalization of Menon–Sury identity to number fields involving a Dirichlet character

Abstract

For every positive integer n, Sita Ramaiah’s identity states that

$$\begin{aligned}&\sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} \gcd (a_1+a_2-1,n) = \phi _2(n)\sigma _0(n) \\&\quad \text { where } \; \phi _2(n)= \sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} 1, \end{aligned}$$

where $(\mathbb {Z}/n\mathbb {Z})^*$ is the multiplicative group of units of the ring $\mathbb {Z}/n\mathbb {Z}$ and $\sigma _s(n) = \displaystyle \sum \nolimits _{d\mid n}d^s$. 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 $\chi$. Our result is a further generalization of recent results in Ji and Wang (Ramanujan J 53:585–594, 2020) and Sury (Rend Circ Mat Palermo 58:99–108, 2009).