## 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