Dedekind zeta function

For a number field $K$ , the Dedekind zeta function, initially on $\operatorname{Re}(s)>1$ , is

\zeta_K(s)=\sum_{0\ne\mathfrak a\subseteq\mathcal O_K}(N\mathfrak a)^{-s}.

The sum runs over all nonzero integral ideals, including the unit ideal, and $N\mathfrak a$ is the absolute ideal norm. The series converges absolutely in this half-plane.

How to read the sum

It is a sum over ideals, not over generators or ideal classes. For $K=\mathbb Q$ , the ideals are $n\mathbb Z$ with $n\ge1$ , giving $\sum_{n\ge1}n^{-s}$ .

At $s=2$ every summand is positive and the value is finite. This is the value used in the Bianchi volume formula; no analytic continuation is needed to interpret it.

References

F. Paulin, Regards croisés sur les séries de Poincaré et leurs applications, Theorem 6, p. 12. Author’s text Definition immediately before Theorem 6.