An application of Birch-Tate formula to tame kernels of real quadratic number fields

Let $F$ be a real quadratic number field with discriminant $D$ and ${\mathcal{O}}_F$ the ring of integers in $F$. Let ${\chi }_F$ be the Dirichlet character associated to $F\mathrm{/}\mathbb{Q}$. Write $L({\chi }_F,s)$ for the Dirichlet L-function of ${\chi }_F$. By an induction argument for imprimitive Dirichlet L-values, we get several $2$-divisibility results on $L({\chi }_F,-1)$ when $D$ has arbitrarily finitely many prime divisors. As an application, by making use of the Birch-Tate formula for $F$, we determine the $2$-primary part for the second $K$ group $K_2{\mathcal{O}}_F$. We also give a new proof for an old theorem of Browkin and Schinzel.