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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 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.

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  • $\begingroup$ Perhaps it would be better to write $o(1)$ instead of $\delta$ in the last display. Because what you mean is: the bound holds for $D$ sufficiently large in terms of $\delta$, and this is exactly what the $o(1)$ notation stands for. $\endgroup$ – GH from MO Oct 23 '16 at 17:18
  • $\begingroup$ @GHfromMO Thanks for the suggestion, I've edited to change the notation. $\endgroup$ – Myshkin Oct 23 '16 at 17:50
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    $\begingroup$ "Much better" depends on the application. The Lavrik bound is still within a constant factor (indeed, within a factor of $2$) of the bound using $h=1$ and Dirichlet, so if 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
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Stephane Louboutin 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 $|L(1,\chi)|$). 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 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).

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