# Residue at $s=1$ for $\zeta$-functions

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?

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There are a great many zeta-functions in mathematics today. I think you are referring to the zeta-function of a number field, the Dedekind zeta-function, but perhaps you could edit your question to include this information explicitly. –  Gerry Myerson Nov 5 '11 at 2:49

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.