TITLE: Residue at $s=1$ for $\zeta$-functions
QUESTION [8 upvotes]: 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?

REPLY [9 votes]: As you anticipated, by estimating the value of $L$-functions at $s=1$, one can obtain bounds on the class group and the regulator. Roughly, one has an inequality $h R < |D|^{1/2 + \epsilon}$, where $h$, $R$, and $D$ are the  class number, regulator, and discriminant of the field respectively. More precisely, one has the Brauer-Siegel theorem.