# Question about a lesser-known "class number formula" of Gauss

My question refers to article 301 of section 5 of Gauss's D. A. - there Gauss gives an asymptotic formula for the mean number of classes of forms with positive discriminant ($$D>0$$):

$$h(D) = \frac{4}{\pi^2}\log (D) + \delta$$

where $$\delta$$ is the following constant:

$$\delta = \frac{8}{\pi^2}C+\frac{48}{\pi^4}\sum_{n=2}^{\infty}\frac{\log (n)}{n^2} - \frac{2 \log 2}{3\pi^2}$$

and $$C$$ is the Euler-Mascheroni constant. This formula is noteworthy because Gauss also evaluates the error term $$\delta$$ in this class formula. I tried to search on the web for references to this analytic result of Gauss but all I found is articles about his asymptotic class number formula for forms with negative discriminant ($$D<0$$), which Gauss gives in the next article of D. A. (article 302).

My question is mainly about getting a reference for this result of Gauss. If such a reference doesn't exist, I'll be glad to hear an expert opinion on the historic significance of this formula (Gauss did publish this formula; thereby it's plausible that it did have influence).

• He gives an asymptotic for the number of genera, which is akin to giving an asymptotic for $\sum_{n\leq X} 2^{\omega(n)}$ —- hence the log. [Note that he says genera around $+D$ or $-D$, and $h(-D)$ is way bigger.] Let me know if I’ve misunderstood! Jan 21 '20 at 1:28
• Mmmh... Isn't art. 301 of the Disquisitiones a reference for this result of Gauß? Jan 21 '20 at 5:35
• @JoséHdz.Stgo. --- I presume with "reference for this result of Gauss", the OP means "a reference which discusses this result of Gauss". Jan 21 '20 at 7:56
• If I recall correctly there may be some information on this in Ayoub's book "Introduction to Analytic Number Theory". Having said this, I have not looked at this book for many years and I do not have easy access to a copy. Jan 21 '20 at 16:42
• @CarloBeenakker: I wonder if the OP is aware of this previous question in MO: mathoverflow.net/a/109396/1593 Jan 22 '20 at 5:05

Suppose, for example, that it's a question of obtaining the mean number of genera for the determinant $$-n$$, a number which we shall denote by $$F(n)$$. If one compares art. 231 (of the D.A) where all the complete characters assignable a priori are enumerated, with art. 261 and 287, where the illustrius author showed that only half of these characters correspond to really existing genera, one could easily find (...) five equations (...) which (on appropriate summing and substituting), result in the asymptotic formula of the mean value of the number of genera for a determinant $$-n$$ as $$\frac{4}{\pi^2}(\mathbb{log (n)}+\frac{12C'}{\pi^2}+2C-\frac{1}{6}\mathbb{log (2)})$$ which coincides with the result of M. Gauss.