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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 \(k\).

MSC:

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
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