On a generalization of Menon-Sury identity to number fields involving a Dirichlet Character

TL;DR

This paper generalizes a classical number theory identity involving gcd sums and multiplicative functions from integers to algebraic number fields, incorporating Dirichlet characters for broader applicability.

Contribution

It extends Menon-Sury type identities to number fields with Dirichlet characters, building on recent related results and broadening the scope of such identities.

Findings

Established a new identity in algebraic number fields involving Dirichlet characters.

Unified previous results and extended their applicability to more general settings.

Provided explicit formulas generalizing classical gcd sum identities.

Abstract

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) = \phi_2(n)\sigma_0(n) \; \text{ where } \; \phi_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 $\sigma_s(n) = \displaystyle\sum_{d\mid n}d^s$. \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 $\chi$. Our result is a further generalization of a recent result in \cite{wj} and \cite{sury}.