Schemes over F1 and zeta functions

We determine the real counting function N(q) (q is an element of [1, infinity)) for the hypothetical 'curve' C = (Spec Z) over bar over F-1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1, infinity) and takes the value -infinity at q = 1 as expected from the infinite genus of C. Then, we develop a theory of functorial F-1-schemes which reconciles the previous attempts by Soule and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adele classes over an arbitrary global field, how to apply our functorial theory of mo-schemes to interpret conceptually the spectral realization of zeros of L-functions.