A generalization of T\'oth identity in $\mathbb{F}_q[T]$ involving a Dirichlet Character

TL;DR

This paper generalizes Toth's identity within the polynomial ring over a finite field, incorporating arithmetical functions and characters to extend its applicability in algebraic number theory.

Contribution

It introduces a broad generalization of Toth's identity in $ ext{F}_q[T]$, involving both multiplicative and additive characters, expanding the theoretical framework.

Findings

Generalized Toth identity in $ ext{F}_q[T]$

Inclusion of arithmetical functions and characters

Potential applications in algebraic number theory

Abstract

Let $\mathbb{A}=\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$. In this article, we prove a generalization of T\'oth identity on $\mathbb{A}$ involving arithmetical functions, multiplicative and additive characters.