Erdos Kac for imaginary class number - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:00:51Z http://mathoverflow.net/feeds/question/88161 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88161/erdos-kac-for-imaginary-class-number Erdos Kac for imaginary class number Will Jagy 2012-02-10T23:07:25Z 2012-02-11T00:55:33Z <p>In answer to <a href="http://mathoverflow.net/questions/41187/a-coverage-question" rel="nofollow">http://mathoverflow.net/questions/41187/a-coverage-question</a> Cam mentions an article by <a href="http://arxiv.org/abs/0707.0237" rel="nofollow">SOUND</a>. I have been running a computer program for <a href="http://mathoverflow.net/questions/88048/on-the-class-number" rel="nofollow">THIS</a> 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 </p> <blockquote> <p>Typically $L(1, \chi_{-d})$ has constant size; rarely does it fall outside the range $(1/10,10).$</p> </blockquote> <p>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. </p> <p>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. </p> <p>So, that is the question, average and variance for the class number of imaginary quadratic fields. </p> <p>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... </p> <p>EDIT: Based on Noam's comment, maybe it is $\log h(-d)$ that has a nice mean and variance.</p> <p>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. </p> http://mathoverflow.net/questions/88161/erdos-kac-for-imaginary-class-number/88168#88168 Answer by Denis Chaperon de LauziÃ¨res for Erdos Kac for imaginary class number Denis Chaperon de LauziÃ¨res 2012-02-11T00:55:33Z 2012-02-11T00:55:33Z <p>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.)</p> <p>See for instance: <a href="http://www.dms.umontreal.ca/%7Eandrew/PDF/L1chi2002.pdf" rel="nofollow">http://www.dms.umontreal.ca/%7Eandrew/PDF/L1chi2002.pdf</a></p>