Trace formula in noncommutative geometry and the Riemann hypothesis. (Formule de trace en géométrie non-commutative et hypothèse de Riemann.) (French) Zbl 0864.46042

Summary: We reduce the Riemann hypothesis for \(L\) functions on a global field \(k\) to the validity (not rigorously justified) of a trace formula for the action of the idele class group on the noncommutative space quotient of the adeles of \(k\) by the multiplicative group of \(k\).


11M55 Relations with noncommutative geometry
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11R56 Adèle rings and groups
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J90 Applications of PDEs on manifolds