Selecta Mathematica

, Volume 5, Issue 1, pp 29-106

Trace formula in noncommutative geometry and the zeros of the Riemann zeta function

  • A. ConnesAffiliated withCollege de France, 3, Rue d'Ulm, F-75005 Paris, France, e-mail:

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We give a spectral interpretation of the critical zeros of the Riemann 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 \( \delta \) of our previous approach.

Key words. Trace formula, zeros, zeta, Adeles