## Did Gauss know Dirichlet’s class number formula in 1801?

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Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below.

In Carl Ludwig Siegel's paper titled The Average Measure of Quadratic Forms With Given Discriminant and Signature Siegel cites two formulae given by Gauss in Disquisitiones Arithmeticae:

(a) $\displaystyle\sum\limits_{d= -N }^1 h_d \sim \frac{\pi}{18 \zeta(3)}N^{3/2}$

(b) $\displaystyle\sum\limits_{d = 1}^N h_d \log{\epsilon}_d \sim \frac{{\pi}^2}{18 \zeta(3)}N^{3/2}$

Where $N > 0$ and $\epsilon_{d} = \frac{1}{2}(t + u \sqrt{d})$ where $(t,u)$ is the smallest positive solution to $t^2 - ud^2 = 4$.

(Actually, Gauss restricts to consideration to binary quadratic forms with even middle coefficient correspondingly arrives at different formulae, but they're essentially the same as those above).

Siegel gives two proofs of these formulae: one proceeding from Dirichlet's class number formula together with character sum estimates due to Polya and Landau, and one via a direct lattice point counting argument.

In light of the facts that (i) I haven't heard anyone say that Gauss's was the one to discover the class number formula and (ii) the character sum estimates seem outside of the scope of Gauss's work, I imagine that his argument was via lattice point counting. Do we have any evidence otherwise? (I checked Gauss's book and he doesn't describe his methods there.)

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A short comment for now: Gauss understood the connection between lattice points on spheres and class numbers of definite quadratic forms: The number of representations of $m$ as a sum of 3 squares is a constant times $h(-m)$ or $h(-4m)$ depending on the congruence of $m$ mod $8$, as I recall. The estimate (a) can probably be deduced from counting lattice points in $R^3$. – Marty Oct 10 at 21:14
@ Marty - aah, good point, I forgot about that result of Gauss. I wonder if there's an analogous result involving class numbers of real quadratic fields. – Jonah Sinick Oct 10 at 22:08

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@ Marty - BTW, Shimura has a great article discussing a common framework for thinking about the ternary quadratic form given by the discriminant and the ternary quadratic form that you mention: ams.org/journals/bull/2006-43-03/… – Jonah Sinick Oct 10 at 22:10
Am I right that it is still not known whether there exists a set of $d>0$ of positive density for which $h_d=1$? I've heard a talk (long time ago) where this was referred to as a "Gauss problem". There is also some nice connection between $h_d$ and the length of the period of the continued fraction expansion of $\sqrt{d}$ but I don't quite remember what it was. – Nikita Sidorov Oct 10 at 23:17

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Jonah is right about rings of integers in number fields, but that's a slightly different question (the $h(5^{2k+1})$ example involves non-maximal orders). – Henry Cohn Oct 11 at 0:29
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## 1 Answer

In 1801, Gauss certainly was aware of the general procedure how to obtain the class number formula (or asymptotic results) via counting lattice points. As a matter of fact, the approach using lattice points in general, and Gauss's circle problem in particular, can already be found in Legendre's Theorie des Nombres in 1798, in connection with his apoproach to the three-squares theorem.

There do exist a couple of posthumous papers by Gauss on this topic, which can be found in his collected works as well as in Maser's German translation of the Disquisitiones (but not, unfortunately, in the English tranlsation). In fact Gauss attempted twice to publish his proof of the class number formula; the first attempt begins with the sentence "33 years have passed since the principles of the wonderful connection, to which this memoir is dedicated, was discovered, as I have remarked at the end of the Disquisitiones". Here Gauss refers to the last paragraph of the Disquisitiones, where he reports to have discovered the analytic solution to a problem stated in arts. 306 and 302. The second version of his manuscript begins with the same sentence, except that the 33 years have been replaced by 36 years.

In any case what this means is that the question in your title should be anwsered with a firm "yes".

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What prevented Gauss from publishing? – Igor Rivin Oct 11 at 18:03
You meant 1798? – Abdelmalek Abdesselam Oct 11 at 18:07
@Igor: probably his famous motto "pauca sed matura"... – Sylvain JULIEN Oct 11 at 19:35
@ Franz Lemmermeyer - Thanks very much! I had never heard of this before. A few follow up questions: (1) When you say that Gauss attempted to publish do you mean that he was somehow unsuccessful in doing so? If so, what was the reason for this? (2) So it looks like he knew the class number formula, but is that what he used to obtain the results that I state above? I would still bet on a lattice point argument in light of the need for character sum estimates if one uses the class number formula approach... – Jonah Sinick Oct 11 at 20:52

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@Igor: lack of time, that is, astronomy, physics, geography etc. – Franz Lemmermeyer Oct 11 at 21:04
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