TITLE: Dirichlet's regulator vs Beilinson's regulator
QUESTION [14 upvotes]: Consider a number field $F$ with ring of integers $O_F$. The Beilinson regulator can in this particular setting be viewed as a map from $K_n(O_F)$ to a suitable real vector space. Here $n$ is any positive odd integer. For $n \geq 3$, there is also another regulator map, defined by Borel, and Burgos has proved that the Borel regulator (suitably normalized) is twice Beilinson's regulator. For $n=1$, the Borel regulator is not defined (as far as I understand, correct me if I'm wrong), but we do have the original and most basic example of a regulator, which is that of Dirichlet.
Question: Is there some form of comparison theorem between the Beilinson regulator and the Dirichlet regulator?

REPLY [2 votes]: As you comment, the Beilinson regulator is defined using Chern classes for Deligne cohomology.  In particular, for $K_1(\mathbb C)$, the first Chern class induces the identity from $K_1(\mathbb C)\simeq \mathbb C^* $ to $H^1_{Deligne}(pt, \mathbb Z(1))\simeq \mathbb C^*$.  One passes to  $H^1_{Deligne}(pt, \mathbb R(1))\simeq \mathbb R$ by taking the logarithm.  So the Belinson regulator for $K_1$ is induced by the map $\log |\; |:K_1(\mathbb C)\to \mathbb R$.