We have an elementary sharp lower bound for the regulator of a real quadratic field as a function of the discriminant

$$R\geq \tfrac{1}{2}(\sqrt{d-4}+\sqrt{d})$$

It is sharp because the equality holds infinitely often for $d=x^2+4$.

The problem of finding a good upper seems much more complicated, but there's still is a very nice (and relatively easy) bound for $\mathbb{Q}(\sqrt{D})$ depending only on $D$.

Loo-Keng Hua proved that $L(1,\chi)<1+\tfrac{1}{2} \log D$, so using the trivial estimate $h\geq 1$ and substituting on Dirichlet's class number formula we get

$$R<\sqrt{D}(\tfrac{1}{2}\log D+1)$$

The way this bound is presented (indirectly) in this survey suggests that it might be the best one currently known for all real quadratic fields (well, for $D>5$). Given how old Hua result is (1942), this seems unlikely, but I haven't been able to find a better one so far.

I am aware of much better estimates which work *for sufficiently large* $D$. For example it follows from work of Lavrik that

$$R<(0.263+\mathcal{o}(1))\sqrt{D}\log D$$

What is the best known bound for $R$ which holds

for all real quadratic fieldsand depends only on $D$? (or for $D>k$ with the $k$ explicit and "small")

I'm also interested in what the true bound is expected to be.

that's the best known then for many purposes it's essentially equivalent to the bound with a coefficient of $1/2$ in place of $0.263$. The tough question is whether the regulator is $o(\sqrt{D} \log D)$ for large $D$, and if so then by how much. $\endgroup$ – Noam D. Elkies Oct 23 '16 at 18:50