Unless I am mistaken, we know that an upper bound for the class number $h(D)$ of a real quadratic field $\mathbb{Q}(\sqrt{D})$ is $O(D^{1/2})$. Is the exponent of $1/2$ known to be the best possible?

Also, is there any better exponent known for the upper bound of $$\liminf_{D \rightarrow \infty} h(D) \, ?$$

Of course, the conjecture is that the exponent shoud be zero, but do we know any better than $1/2$?


The exponent $1/2$ is best possible. You can see this by varying $D$ along values of the form $n^2+4$ so that the regulator is only of size about $\log D$. Then the lower bounds for $L(1,\chi)$ (Siegel's theorem) and the class number formula give such a lower bound. This was worked out more precisely by Montgomery and Weinberger; see this paper of Duke which works out analogs for other number fields and gives the history: http://www.math.ucla.edu/~wdduke/preprints/number.pdf

Regarding the other question, I don't think we know much more than the regulator being larger than a power of $(\log D)$ infinitely often. So there is a big gap here. See the paper of Jacobson, Lukes and Williams on computational results towards this problem and which also discusses the known theoretical results: http://citeseerx.ist.psu.edu/viewdoc/download?doi=

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