A generalization of Arai-Carlitz's identity (Q829846)

The authors of the paper present a generalization of the Arai-Carlitz's identity to the ring of algebraic integers concerning Dirichlet characters. The formulation of the general assertion is quite complex, but the general assertion implies the following interesting equality \[ \sum\limits_{a,b,a+b\in(\mathbb{Z}/n\mathbb{Z})^\times}\mathrm{gcd}(a+b-1,n)\chi(a)=\mu(d)\varphi(n_0^2/d)X(n/n_0)\tau(n/d) \] provided if \(n\) is positive integer and \(\chi\) is a Dirichlet character modulo \(n\) with conductor \(d\). Here \(n_0|n\) is such that \(n_0\) has the same prime factors with \(d\) and \(\mathrm{gcd}(n_0,n/n_0)=1\), \(\varphi\) denotes the Euler's totient function, \(\mu\) is the Möbius function, \(\tau\) is the divisor function, and \[ X(n)=\sum\limits_{a,b,a+b\in(\mathbb{Z}/n\mathbb{Z})^\times}1. \]