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

Connes, Alain

C. R. Acad. Sci., Paris, Sér. I 323, No. 12, 1231-1236 (1996).

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

Keywords:

Riemann hypothesis; \(L\) functions; trace formula; idele class group; noncommutative space; adeles