# Erdos Kac for imaginary class number

In answer to A coverage question Cam mentions an article by SOUND. I have been running a computer program for THIS and would like to know if there are a reasonable average and standard deviation for the class number, related to Dirichlet's formula for $d > 4$ $$h(-d) = \sqrt d \; L(1, \chi_{-d}) \; / \; \pi.$$ Sound writes

Typically $L(1, \chi_{-d})$ has constant size; rarely does it fall outside the range $(1/10,10).$

This suggests a possible calculation of standard deviation, as I am seeing articles about "second moments" of the zeta function and $L$-functions, although nothing I can interpret.

Note: The original Erdos-Kac may be of an entirely different nature; it says that, in the long run, the number of prime divisors of a number $n$ is normally distributed with mean $\log \log n$ and standard deviation $\sqrt{ \log \log n},$ this being the colloquial description of a precise statement.

So, that is the question, average and variance for the class number of imaginary quadratic fields.

P. S. The computer program I am running is restricted to the above with $d \equiv 3 \pmod 4,$ but does not rule out square factors of $d$ ahead of time. In the first occurrence of such $d$ with a target class number, $d$ is almost always squarefree. Indeed, with class numbers up to 4000, the only exception is class number 104, which first occurs at $d= 9359 = 7^2 \cdot 191.$ If that issue matters, I would be delighted to hear about it...

EDIT: Based on Noam's comment, maybe it is $\log h(-d)$ that has a nice mean and variance.

EDIT ANOTHER: the most interesting case is $d \equiv 3 \pmod 4$ where $d$ is prime. Noam had pointed out in one of the threads that primality is required to achieve an odd class number.

• I'd guess that it's $\log L(1,\chi_{-d})$ that would be nicely distributed, since it's morally the sum of independent contributions $-\log(1-\chi_{-d}(p)/p)$. – Noam D. Elkies Feb 10 '12 at 23:29
• @Noam, thanks. Sound's phrasing suggested a logarithm. Since you bring it up, the class number formula itself does say that $L(1,\chi_{-d})$ is positive, not something that had occurred to me. – Will Jagy Feb 10 '12 at 23:38

## 1 Answer

There are many papers about strong probabilistic models of $L(1,\chi_d)$, in particular by Granville and Soundararajan. These are quite precise (basically, because the Euler product at 1 is "almost" absolutely convergent, one can model its value by a random Euler product, and even prove that this model is close to the truth when taking discriminants of bounded size.)

See for instance: http://www.dms.umontreal.ca/%7Eandrew/PDF/L1chi2002.pdf

• Thanks, Denis. Let me see what I can understand of this article. – Will Jagy Feb 11 '12 at 1:14