Remark on a Simple Proof of the Mean Value of $K_2(\mathcal{O})$ in Function Fields

Let ${\mathbb{F}}_q$ denote a finite field of odd cardinality $q$, $\mathbb{A}={\mathbb{F}}_q[T]$ the polynomial ring over ${\mathbb{F}}_q$ and $k={\mathbb{F}}_q(T)$ the rational function field over ${\mathbb{F}}_q$. In this paper, we compute the average value of the size of the group $K_2({\mathcal{O}}_{\gamma D})$, where ${\mathcal{O}}_{\gamma D}$ denotes the integral closure of $\mathbb{A}$ in $k(\sqrt{\gamma D})$, $D$ is a monic, square-free polynomial of even degree and $\gamma$ is a fixed generator of ${\mathbb{F}}_q^{\ast }$.