Is there any sort of a bound for the magnitude of this residue? I've been looking at some algebraic number theory problems from Princeton's general exam. One of them is the following:
Why does a ''large'' fundamental unit suggest a ''small'' class number and vice versa?
To me it seems like this has to have something to do with the relation given by the residue of the zeta function. A ''large'' fundamental unit would imply a large regulator, but I can't see why this would force the class number to be small unless we can bound the magnitude of the residue. Any ideas?