Zeta Spectral Triples
Alain Connes, Caterina Consani, Henri Moscovici

TL;DR
This paper introduces a spectral approach to the Riemann Hypothesis by constructing self-adjoint operators whose spectra closely match the non-trivial zeros of the zeta function, supported by numerical evidence and theoretical foundations.
Contribution
It develops a novel spectral realization of zeta zeros using rank-one perturbations of spectral triples, linking number theory with operator theory and providing numerical evidence for the hypothesis.
Findings
Operators' spectra match zeta zeros with high accuracy
Spectra converge to zeros as parameters grow
Regularized determinants relate to the Riemann Xi function
Abstract
We propose and investigate a strategy toward a proof of the Riemann Hypothesis based on a spectral realization of its non-trivial zeros. Our approach constructs self-adjoint operators obtained as rank-one perturbations of the spectral triple associated with the scaling operator on the interval . The construction only involves the Euler products over the primes and produces self-adjoint operators whose spectra coincide, with striking numerical accuracy, with the lowest non-trivial zeros of , even for small values of . The theoretical foundation rests on the framework introduced in "Spectral triples and zeta-cycles" (Enseign. Math. 69 (2023), no. 1-2, 93-148), together with the extension in "Quadratic Forms, Real Zeros and Echoes of the Spectral Action" (Commun. Math. Phys. (2025)) of the classical Caratheodory-Fejer…
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Taxonomy
TopicsSpectral Theory in Mathematical Physics · Random Matrices and Applications · Mathematical functions and polynomials
