A generalization of Tóth identity in the ring of algebraic integers involving a Dirichlet Character

Abstract

The $k$-dimensional generalized Euler function ${\phi }_k(n)$ is defined to be the number of ordered $k$-tuples $(a_1,a_2,\dots ,a_k)\in {\mathbb{N}}^k$ with $1\le a_1,a_2,\dots ,a_k\le n$ such that both the product $a_1{a}_2\cdots a_k$ and the sum $a_1+a_2+\cdots +a_k$ are co-prime to $n$. Tóth proved that the identity \begin{equation*} \sum_{\substack{a_1,a_2,\ldots, a_k=1 \\ \gcd(a_1a_2\cdots a_k,n)=1\\ \gcd(a_1+a_2+\cdots+a_k,n)=1}}^n \gcd(a_1+a_2+\cdots+a_k-1,n) =φ_k(n)σ_0(n), \;\; \text{ where } σ_s(n) = \sum_{d\mid n}d^s \;\; \text{ holds. } \end{equation*} This identity can also be viewed as a generalization of Menon's identity. In this article, we generalize this identity to the ring of algebraic integers involving arithmetical functions and Dirichlet characters.