The Two-Dimensional Distribution of Values of ζ(1 + it)

We prove several results on the distribution function of ζ(1 + it) in the complex plane, that is, the joint distribution function of arg ζ(1 + it) and |ζ(1 + it)|. Similar results are also given for L(1, χ) (as χ varies over nonprincipal characters modulo a large prime q).