A generalization of Arai–Carlitz’s identity

Abstract

Let n be a positive integer. Arai-Carlitz’s identity is

$$\begin{aligned} \sum _{a, b, a+b\in ({\mathbb {Z}}/n{\mathbb {Z}})^{\times }}\gcd (a+b-1, n)=X(n)\tau (n), \end{aligned}$$

where $X(n)=\sum _{a, b, a+b\in ({\mathbb {Z}}/n{\mathbb {Z}})^{\times }}1$ and $\tau (n)$ is the divisor function. In this paper, we generalize Arai-Carlitz’s identity to the ring of algebraic integers concerning Dirichlet characters. In particular, let n be a positive integer and $\chi$ a Dirichlet character modulo n with the conductor d. Then

$$\begin{aligned} \sum _{a, b, a+b\in ({\mathbb {Z}}/n{\mathbb {Z}})^{\times }} \gcd (a+b-1, n)\chi (a) = \mu (d)\varphi (n_{0}^{2}/d)X(n/n_{0})\tau (n/d), \end{aligned}$$

where $n_{0}|n$ such that $n_{0}$ has the same prime factors with d and $\gcd (n_{0}, \frac{n}{n_{0}})=1$, $\varphi (n)$ is the Euler’s totient function and $\mu (n)$ is the Möbius function.