Zeta spectral triples
Alain Connes, Caterina Consani, Henri Moscovici
Abstract
We propose and investigate a strategy towards a proof of the Riemann hypothesis based on a spectral realization of its non-trivial zeros. Our approach constructs self-adjoint operators $D\_{\log}^{(\lambda,N)}$ obtained as rank-one perturbations of the spectral triple associated with the scaling operator on the interval $\[\lambda^{-1}, \lambda]$. The construction only involves the Euler products over the primes $p \leq x = \lambda^{2}$ and produces self-adjoint operators whose spectra coincide, with striking numerical accuracy, with the lowest non-trivial zeros of $\zeta(\frac{1}{2} + i s)$, even for small values of $x$. The theoretical foundation rests on the framework introduced by Connes and Consani (2023) together with the extension by Connes and van Suijlekom (2025) of the classical Carathéodory–Fejér theorem for Toeplitz matrices, which guarantees the necessary self-adjointness. Numerical experiments show that the spectra of the operators $D\_{\log}^{(\lambda,N)}$ converge towards the zeros of $\zeta(\frac{1}{2}+ i s)$ as the parameters $N, \lambda \rightarrow \infty$. A rigorous proof of this convergence would establish the Riemann hypothesis. We further compute the regularized determinants $\det\_{\mathrm{reg}}(D\_{\log}^{(\lambda,N)}- z)$ of these operators and discuss the analytic role they play in controlling and potentially proving the above result by showing that, suitably normalized, they converge towards the Riemann $\Xi$ function.