Let $N$ be a positive integer. Denote by $\varepsilon(N)$ to be the number of fundamental units which appear for some real quadratic field. The goal is to prove that $\varepsilon(N)$ is asymptotic to $2N$.
Here is an incomplete argument that I have thus far.
If $\varepsilon$ is a fundamental unit of $L = \mathbb{Q}(\sqrt{d})$ where $d>1$ and square-free then $\varepsilon = a + b \sqrt{d}$ where $a,b$ are positive integers or positive half-integers (depending on $d$). If also require that $1 \leq \varepsilon \leq N$ then we see that $a\leq N$. It follows from here $\text{tr}_{L/\mathbb{Q}}(\varepsilon) = 2a$ and $N_{L/\mathbb{Q}}(\varepsilon) = \pm 1$. Thus, the minimal polynomial for $\varepsilon$ is given by $X^2 – cX \pm 1$ where $1\leq c \leq 2N$.
The larger root of this minimal polynomial is given by $c + \sqrt{c^2 \pm 4}$. If $c > N+1$ then this larger root is $>N$ and falls outside the allowable interval. Thus, we conclude that the fundamental units in the interval $[1,N]$ have minimal polynomial given by $X^2 – cX \pm 1$ where $1\leq c\leq N+1$.
We have a total of $2(N+1)$ such polynomials. The fundamental unit will be the larger root (by definition of fundamental unit in a real quadratic field). Thus, we have at most $2(N+1)$ such roots in the interval $[1,N]$. Of course, some of those roots might be units which are not fundamental, however, this is just an upper bound, and so we have,
$$ \varepsilon(N) \leq 2(N+1) $$
Now we need some lower bound estimate. I have been trying to do several attempts but they are not working out so far. My intuition tells me the lower bound estimate should look something like $2N – \sqrt{N}$. But I cannot find any argument for that as I do not see how one can tell if such a polynomial has a fundamental unit as its root.
I was trying to estimate those $c$ for which $c^2 \pm 4$ is square-free. Those such $c$ will definitely lead to polynomials which have a root as a fundamental unit. But it is not clear to me how to estimate that quantity.
Best Answer
With $U$ the units of every ring of integers of real quadratic fields let $$r(y) = \sum_{u \in U, u \in (1,y]} 1 = \sum_{X^2-cX+1, c \ge 3,u = \frac{c+\sqrt{c^2-4}}{2} \le y} 1+\sum_{X^2-cX-1, c \ge 1, u = \frac{c+\sqrt{c^2+4}}{2} \le y} 1 \\= 2y+O(1)$$ with $F$ the fundamental units of every ring of integers of real quadratic fields let $$\varepsilon(y) =\sum_{u \in F, u \in (1,y]} 1$$ since $$O_{\mathbb{Q}(\sqrt{d})}^\times = \{1,-1\} \times u_d^\mathbb{Z}, \qquad O_{\mathbb{Q}(\sqrt{d})}^\times \cap (1,y] = \{ u_d^k, k \in 1 \ldots\frac{\log y}{\log u_d}\}$$ we have $$r(y) = \sum_{k \ge 1}\sum_{u \in F, u^k \in (1,y]} 1 = \sum_{k \ge 1}\varepsilon(y^{1/k})$$ and by Möbius inversion
$$\varepsilon(y)= \sum_{k \ge 1} \mu(k) r(y^{1/k}) = \sum_{k \ge 1} (\mu(k)2y^{1/k}+O(1))= 2y- 2y^{1/2}+O(2y^{1/3}\log_{3/2} y)$$