Sorry sorry, I meant genus at least 1, Siegel's theorem covering 1. The "almost all" takes out the singular hypersurfaces. I believe some very easy sieve theory can prove this since, mod p, most assignments of the other coefficients should give a squarefree polynomial.

You can add the preamble into jsMath (in the macros section I believe). Bandwidth is not an issue - the browser downloads the library practically only once. And finally, making a Firefox addon that does this is simple, and I take on the challenge to do it during passover.

I suspect there is a uniform bound on such r: given an r, there is the hypersurface $D(b_0,...,b_{n-1})/D_0 = y^2$, where $D$ is the discriminant of the monic polynomial, and $D_0$ will be the discriminant of a fixed monic polynomial. Not all solutions to this equation will be in the same field, but probably infinitely many. I believe there are conjectures on the number of integer solutions to hypersurfaces in $r$ variables, can anyone who knows shed more light?

Would Satoh's algorithm for computing zeta functions, that relies on Dwork's work on deformations, be related to p-adic analysis? It's different than Kedlaya's work, but still uses p-adic geometry stuff that I don't understand.

Hmmm. Is this right: I took all monic degree 3 polynomials with coefficients abs value up to 100, with square discriminant that defined different fields. Then computed I the geometric mean of $4 h_K R_k/\sqrt{D_k}$, where K is each such cyclic field. This turned out less than 1. Then I order the fields by discriminant and took partial means, but it remains less than 1. At 100, the value is 0.6546.

It does! Lets check for cubics now, and if it works, then this must be what Scholz meant! Also, about the large prime factors condition, is it necessary? The data seemed to suggest it doesn't.

@moonface: I just checked in sage and got: $(\prod_{0 < -D < X} h(D))^{1/r(X)} \sim c\sqrt{X}$ , where $r(X)$ is the number of fundamental negative discriminants greater than $−X$ , and the sum ranges only on those as well. The experimental data says that $c$ is about 0.231, not close to 6. Did I misunderstand?

I don't think the question needs clarification, nor do all examples exist on Wikipedia.

"A" being Gauss in Jacobi's quote.

@Georges: This is funny, your comment is getting more votes than the question itself - while people agree with you, they still really don't care for game theory.

The downvotes are encouragement to study a more popular subject. This has a lot of sense behind it. The users of Math Overflow want Soheil to get a good job at a good university surrounded by other good researchers. Maybe if we all downvoted every question that wasn't close enough - Riemann's hypothesis would be solved by now!

@Hansen: That is a, and is the mentioned, Tauberian theorem.

The Tauberian theorem you mention was actually proved by Raikov: "Generalisation of a theorem of Ikehara–Landau (in Russian). Mat. Sbornik, 45 (1938)". If I am not mistaken, the Selberg-Delange method is more about getting a better error term (dating 1947?). Of course the particular instance of Landau was also proved and can be found in: "Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate." Arch. Math. Phys. 13, 305-312, 1908.

I might be wrong, but I believe this idea predates Selberg and Delange by almost half a century. This was done already in 1908 by Landau to find the number of numbers represented by a sum of two squares. But since Selberg-Delange has been mentioned quite a few times in this post, I am doubting myself.

Even though it is very elementary, I think you should write the above comment in the actual answer. The way it is currently written is just annoying.