For a given real quadratic field $K$, the group of units of its ring of integers is $\mathcal{O}_K^{\times}\cong(\pm1)\times \mathbb{Z}$ by the Dirichlet unit theorem. For each $\mathcal{O}_K$, pick the fundamental unit as $\epsilon >1$, then $\epsilon=m\sqrt{d}+n$, where $m,n>0$ are integers or half integers. Now for a large variable $x>>1$, define the counting function $$ u(x):=\sum_{1<\epsilon<x} 1, $$ where the sum is over all real quadratic fields.

Firstly the sum is finite, since by the above token, there are only finitely many $\epsilon=m\sqrt{d}+n$ with bounded absolute value as $d\rightarrow\infty$.

What asymptotic information is known about $u(x)$?