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

For every positive integer n, Sita Ramaiah’s identity states that $$\begin{aligned}&\sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} \gcd (a_1+a_2-1,n) = \phi _2(n)\sigma _0(n) \\&\quad \text { where } \; _2(n)= \sum 1, \end{aligned}$$ $$(\mathbb {Z})^*$$ is the multiplicative group of units ring $$\mathbb {Z}$$ and $$\sigma _s(n) \displaystyle \nolimits _{d\mid n}d^s$$ . This can also be viewed as a generalization Menon’s identity. In this article, we generalize to an algebraic number field K involving Dirichlet character $$\chi $$ Our result further recent results in Ji Wang (Ramanujan J 53:585–594, 2020) Sury (Rend Circ Mat Palermo 58:99–108, 2009).