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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 ($D>0$):

$$h(D) = \frac{4}{\pi^2}\log (D) + \frac{8}{\pi^2}C+\frac{48}{\pi^4}\sum\frac{\log (n)}{n^2} - \frac{2 \log 2}{3\pi^2}$$

where $C$ is the Euler-Mascheroni constant. This formula is noteworthy because Gauss also tries to evaluate the error term 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).

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  • $\begingroup$ 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! $\endgroup$ – alpoge Jan 21 at 1:28
  • $\begingroup$ Mmmh... Isn't art. 301 of the Disquisitiones a reference for this result of Gauß? $\endgroup$ – José Hdz. Stgo. Jan 21 at 5:35
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    $\begingroup$ @JoséHdz.Stgo. --- I presume with "reference for this result of Gauss", the OP means "a reference which discusses this result of Gauss". $\endgroup$ – Carlo Beenakker Jan 21 at 7:56
  • $\begingroup$ 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. $\endgroup$ – M. Khan Jan 21 at 16:42
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    $\begingroup$ @CarloBeenakker: I wonder if the OP is aware of this previous question in MO: mathoverflow.net/a/109396/1593 $\endgroup$ – José Hdz. Stgo. Jan 22 at 5:05
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Concerning the historic significance of Gauss's article 301: Marius Overholt traces back to this publication on the growth of class numbers the use of local averaging to study the growth of arithmetic functions.

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