Erdos Kac for imaginary class number

2012-02-10T23:07:25Z

In answer to http://mathoverflow.net/questions/41187/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.

Answer by Denis Chaperon de Lauzières for Erdos Kac for imaginary class number

2012-02-11T00:55:33Z

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

Erdos Kac for imaginary class number - MathOverflow

Erdos Kac for imaginary class number

Answer by Denis Chaperon de Lauzières for Erdos Kac for imaginary class number