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For what number fields $K$ can we actually compute the residue of $\zeta_K(s)$ at the pole $s=1$ directly? Since the class number formula tells us that

$$\textrm{Res}_{s=1}\zeta_K(s)=\frac{2^r(2\pi)^s\textrm{reg}(K)h_K}{\# \mu(K)\sqrt{|d_K|}}$$

this doesn't seem terribly useful unless one can calculate everything except one element. The only examples that I've ever seen involve computing the residue, but I don't think that's too interesting as I would expect that the whole utility of the formula comes from computing the residue and then using whatever we know to deduce either the regulator, class number or discriminant.

The only other utility of the formula that I can come up with is if we have some relationship between the zeta-functions of some number fields. Then one could use the functional equation to compute relations between the regulators, class numbers etc. of these number fields. Picking e.g. an $S_3$ extension we find relationships between the zeta functions of the subextensions.

Could anyone elaborate on if this equation actually has any use in practical computations? Even references to papers where its used to compute something interesting would be appreciated.

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BTW, I'm pretty sure that the general answer is "it depends". Since if one could figure out the residue even for quadratic fields, then we could solve the class number problem... –  pki Jan 16 '12 at 21:11

1 Answer 1

For any given field, you can compute the residue of the zeta function to arbitrary precision: just take the product of sufficiently many Euler factors. In practice, this can be quite slow for high degree fields, since you might need lots of factors to get the desired precision.

Of course, computations can never prove the class number 1 problem, since a computation can only check finitely many fields.

The main interest of the formula is that it provides a prototypical example of a "local-global" principle: the left hand side is something that you compute locally at each prime, while the right hand side is something manifestly global. As you probably know, the class number formula has served as a blue print for very deep and far reaching generalisations, such as the Birch and Swinnerton-Dyer conjecture for example. Most of these generalisations are wide open.

Since you also asked for references, e.g. about extracting information about fields from relations between zeta functions, here is a paper of mine in which concrete information about the Galois module structure of the units is extracted from the relationship you mention.

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