  We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for
the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose
corresponding zeta function is the complete Riemann zeta function. Then, we
develop a theory of functorial $\F_1$-schemes which reconciles the previous
attempts by C. Soul\'e and A. Deitmar. Our construction fits with the geometry
of monoids of K. Kato, is no longer limited to toric varieties and it covers
the case of schemes associated to Chevalley groups. Finally we show, using the
monoid of ad\`ele 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.
