The Analytic Class Number Formula This is a slightly soft question. I'm curious about a statement that I've heard off-hand many times that the Analytic Class Number Formula is not really a formula for the class number. I have always interpreted this to mean that even though it is an expression containing the class number, perhaps there are other quantities in the expression that are just as hard to calculate as the class number, and so it is not so much help on that front.
Just for convenience, the statement is that for a number field $F$,
$$
\lim_{s\rightarrow1}(s-1)\zeta_F(s)=\frac{2^r(2\pi)^sh_FR_F}{w_F|d_F|^{1/2}},
$$ 
where the quantities are the usual ones, specifically $R_F$ is the regulator of $F$ and $w_F$ is the number of roots of unity in $F$.
At this point I was wondering if anyone knows particular cases where two of the three quantities:


*

*The residue of $\zeta_F$ at 1,

*The regulator $R_F$,

*The class number $h_F$,


can be calculated, leading to a formula for the remaining one? I would be interested to see if it is always the same two that can be computed, or whether difference situations use the formula in difference directions.
Thanks in advance!
 A: The analytic class number formula really can be used to calculate class numbers in many examples. Often the class number is the most difficult quantity in the formula.
To give an elemenatry example, let $K=\Bbb Q(\sqrt{d})$ be an imaginary quadratic number field with discriminant $d_K$ and class number $h$, and $w$ the number of roots of unitiy in $K$, i.e., $w=2,4,6$, then the formula gives
$$
h=\frac{w\sqrt{|d_K|}}{2\pi}L(1,\chi),
$$
where $\chi(n)=(d_K/n)$ is the quadratic Dirichlet character.
Example: $K=\Bbb Q(\sqrt{-15})$. Then $w=2$, $d_K=-15$ and $L(1,\chi)=\frac{2\pi}{\sqrt{15}}\sim 1.62231147$, so that
$$
h=\frac{\sqrt{15}}{\pi}L(1,\chi)=2.
$$
A: Henri Cohen has written whole books about this question (Computational algebraic number theory). In addition, there is the theorem of Brauer-Siegel which gives you an idea how the "class number" formula can be used. Finally the Brauer-Kuroda relations allow you to compute the class number of a Galois number field from the class numbers (and a unit index) of certain subfields.
This being said, there are a few cases where it can be evaluated more or less explicitly. One such case is that of CM-fields: these are totally complex quadratic extensions of totally real number fields (this includes complex quadratic fields or cyclotomic fields). In this case, the units essentially come from the totally real subfield, and by applying the class number formula to both fields you get a formula for the "relative class number" (in the complex quadratic field this is just the class number since the maximal real subfield is ${\mathbb Q}$ and has class number $1$). St\'ephane Louboutin has written a host of papers on exploiting this situation for the classification of CM-fields with class number $1$.
There are also families of number fields with parametrized units; if $m = r^2+1$, for example, then $\varepsilon = r + \sqrt{m}$ is a unit of the real quadratic number field ${\mathbb Q}(\sqrt{m})$, and except for finitely many cases this unit will be fundamental. Thus for such families you get the regulator for free (or at least an upper bound). Strictly speaking real cyclotomic fields belong to this class since you can write down an independent system of units, the cyclotomic units, but in this case it is very difficult to check whether this system is fundamental, or to compute the index in the full unit group. 
The residue of the zeta function at $s = 1$ can, in principle, always be computed from the Euler product, but for getting good approximations you need to take many primes into account, which becomes prohibitive for number fields with large discriminant.
