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](http://projecteuclid.org/euclid.em/1062621079) 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 fields** and 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.



Upper bounds for regulators of real quadratic fields
[Stephane Louboutin](http://iml.univ-mrs.fr/~loubouti/) has several papers on getting explicit bounds for $L(1,\chi)$, for $\chi$ a character $\pmod q$. They're all of the strength of $1/2 \log q + $ an explicit constant. Some of his results include information on $\chi(2)$ which is sometimes helpful. 


The best theoretical upper bound is due to P.J. Stephens: it gives 
$$
L(1,\chi) \le \frac{1}{4} \Big( 2- \frac{2}{\sqrt{e}} + o(1)\Big) \log q = (0.1967\ldots +o(1)) \log q,
$$ 
for a quadratic character $\chi \pmod q$ (see for example [Upper bounds for $|L(1,\chi)|$](https://arxiv.org/abs/math/0106176)). Here the $1/4$ is from Burgess and the $2-2/\sqrt{e}$ comes from Vinogradov's trick for the least quadratic non-residue. To go beyond $1/2$ in the bound for $L(1,\chi)$ explicitly, one would have to work with either explicit Burgess type bounds or with explicit versions of Vinogradov's trick. I don't think anyone has carried that out -- nothing new here, but just needs elbow grease. 


On GRH Theorem 1.5 from [Lamzouri-Li-Soundararajan](https://arxiv.org/pdf/1309.3595v2.pdf) will give you explicit bounds for $L(1,\chi)$. These show that the regulator is bounded by something of size $\sqrt{d} \log \log d$, which is likely the correct order of magnitude (but this is unknown). 


