  One of the driving motivations to develop $\F_1$-geometry is the hope to
translate Weil's proof of the Riemann hypothesis from positive characteristics
to number fields, which might result in a proof of the classical Riemann
hypothesis. The underlying idea is that the spectrum of $\Z$ should find an
interpretation as a curve over $\F_1$, which has a completion $\bar{\Spec\Z}$
analogous to a curve over a finite field. The hope is that intersection theory
for divisors on the arithmetic surface $\bar{\Spec\Z} \times \bar{\Spec\Z}$
will allow to mimic Weil's proof.
  It turns out that it is possible to define an object $\bar{\Spec\Z}$ from the
viewpoint of blueprints that has certain properties, which come close to the
properties of its analogs in positive characteristic. This shall be explained
in the following note, which is a summary of a talk given at the Max Planck
Institute in March, 2012.

Algebraic Geometry;Number Theory