Blueprints — towards absolute arithmetic?

One of the driving motivations for F1F1-geometry is the hope to translate Weil's proof of the Riemann hypothesis from positive characteristics to number fields. The spectrum of ZZ should find an interpretation as a curve over F1F1, together with a completion SpecZ¯. Intersection theory for divisors on the arithmetic surface SpecZ¯×SpecZ¯ should allow us to mimic Weil's proof. It is possible to define SpecZ¯ as a locally blueprinted space, which shares certain properties with its analog in positive characteristic. In particular, the arithmetic surface SpecZ¯×SpecZ¯ is two-dimensional. We describe the local factors (including the Γ  -factor) of the Riemann zeta function as integrals over the space of ideals of the stalks of the structure sheaf of SpecZ¯. A comparison of line bundles on SpecZ¯ with Arakelov divisor exhibits a second integral formula for the Riemann zeta function. We conclude this note with some remarks on étale cohomology for SpecZ¯.