Trace Formula in Noncommutative Geometry andthe Zeros of the Riemann

We give a spectral interpretation of the critical zeros of the Rie-mann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of Adele classes. This reduces the Riemann hypothesis to the validity of the trace formula and eliminates the parameter of our previous approach. Introduction. I Quantum chaos and the hypothetical Riemann ow. II Algebraic Geometry and global elds of non zero characteristic. III Spectral interpretation of critical zeros. IV The distribution trace formula for ows on manifolds.