Alain Connes
(IHÉS/Collège de France)
June 2, 2021
Organized by Americas GNCG Seminar.

Spectral Triples and Zeta-Cycles

This is joint work with C. Consani. When contemplating the low lying zeros of the Riemann zeta function one is tempted to speculate that they may form the spectrum of an operator of the form 1/2+iD with D self-adjoint, and to search for the geometry provided by a spectral triple for which D is the Dirac operator. We give the construction, using prolate spheroidal functions, of a spectral triple which is a finite rank perturbation of the spectral triple of the circle of length L and admits a spectrum of 1/2+iD very similar to the low lying zeros of the Riemann zeta function. We justify conceptually this result and show that, for each eigenvalue, the coincidence is perfect for the special values of the length L of the circle for which the two natural ways of realizing the perturbation give the same eigenvalue. We test this fact numerically by reproducing the first thirty one zeros of the Riemann zeta function from our spectral side, and estimate the probability of having obtained this agreement at random, as a very small number whose first fifty decimal places are all zero. The theoretical concept which emerges is that of zeta cycle and our main result establishes its relation with the critical zeros of the Riemann zeta function and with the spectral realization of these zeros described in 1999 by the speaker.

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