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

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.)

• 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 $R^3$. Oct 10, 2012 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. Oct 10, 2012 at 22:08
• @ 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/S0273-0979-06-01107-4 Oct 10, 2012 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. Oct 10, 2012 at 23:17
• 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). Oct 11, 2012 at 0:29