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Zeta Spectral Triples

Alain Connes, Caterina Consani, Henri Moscovici

TL;DR

This work advances a spectral-analytic program to tackle the Riemann Hypothesis by realizing nontrivial zeta zeros as spectra of selfadjoint operators built from rank-one perturbations of a scaling-operator, organized through infrared spectral triples and Weil's explicit-form quadratic form. It leverages restricted Euler products, a precise matrix encoding of the Weil form in a finite basis, and truncation tools to define a family of perturbed Dirac-type operators whose spectra track $\zeta(\tfrac12+is)$ with remarkable numerical accuracy. A rigorous convergence pathway is proposed via the regularized determinants tending to the Riemann $\Xi$ function, together with a program to prove key spectral properties (simplicity and parity) and to relate the minimal eigenfunction to prolate-wave analogues. The numerical results provide strong evidence for the convergence mechanism and motivate further analytic work to close the missing steps, potentially yielding a route to RH grounded in noncommutative geometry and information-theoretic perspectives.

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 $[λ^{-1}, λ]$. The construction only involves the Euler products over the primes $p \leq x = λ^2$ and produces self-adjoint operators whose spectra coincide, with striking numerical accuracy, with the lowest non-trivial zeros of $ζ(1/2 + i s)$, even for small values of $x$. 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 theorem for Toeplitz matrices, which guarantees the necessary self-adjointness. Numerical experiments show that the spectra of the operators converge towards the zeros of $ζ(1/2 + i s)$ as the parameters $N, λ\to \infty$. A rigorous proof of this convergence would establish the Riemann Hypothesis. We further compute the regularized determinants 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 $Ξ$ function.

Zeta Spectral Triples

TL;DR

This work advances a spectral-analytic program to tackle the Riemann Hypothesis by realizing nontrivial zeta zeros as spectra of selfadjoint operators built from rank-one perturbations of a scaling-operator, organized through infrared spectral triples and Weil's explicit-form quadratic form. It leverages restricted Euler products, a precise matrix encoding of the Weil form in a finite basis, and truncation tools to define a family of perturbed Dirac-type operators whose spectra track with remarkable numerical accuracy. A rigorous convergence pathway is proposed via the regularized determinants tending to the Riemann function, together with a program to prove key spectral properties (simplicity and parity) and to relate the minimal eigenfunction to prolate-wave analogues. The numerical results provide strong evidence for the convergence mechanism and motivate further analytic work to close the missing steps, potentially yielding a route to RH grounded in noncommutative geometry and information-theoretic perspectives.

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 theorem for Toeplitz matrices, which guarantees the necessary self-adjointness. Numerical experiments show that the spectra of the operators converge towards the zeros of as the parameters . A rigorous proof of this convergence would establish the Riemann Hypothesis. We further compute the regularized determinants 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 function.

Paper Structure

This paper contains 21 sections, 26 theorems, 190 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Banach algebra $L^1(\mathbb{R},dx)$
  4. The basis $\{U_n\}_{n \in \mathbb{Z}}$
  5. The Weil quadratic form $QW$
  6. The quadratic form $QW_\lambda$
  7. Discrete spectrum of the semilocal Weil quadratic form QWlambda
  8. The matrix of $QW_\lambda$ in the basis $V_n$
  9. The matrix $W_{0,2}(V_n,V_m)$
  10. The matrix $W_p(V_n,V_m)$
  11. The matrix $W_\mathbb{R}(V_n,V_m)$
  12. The infrared spectral triples
  13. Truncation of $QW_\lambda$
  14. Properties of truncated matrices
  15. The Dirichlet Kernel $\delta_N$ as an approximation of the Dirac Delta
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\epsilon_N$ be the smallest eigenvalue of $QW_\lambda^N$ assumed simple and $\xi$ the corresponding eigenvector assumed even, normalized by $\delta_N(\xi)=1$. $(i)$ The operator ${D_{\log}^{(\lambda,N)}}=D_{\log}^{(\lambda)}-\vert D_{\log}^{(\lambda)}\xi\rangle\langle \delta_N\vert$ is selfadjo

Figures (4)

  • Figure 1: This shows the differences between the first twenty zeros of $\zeta\left(\frac{1}{2}+i s\right)$ and the eigenvalues of the operator ${D_{\log}^{(\lambda,N)}}$ for $\lambda=3$ and $N=120$.
  • Figure 2: Graph of $e_0(\mu)$ for $\mu\leq 36$.
  • Figure 3: Graph of $e_4(\mu)$ for $\mu\leq 36$.
  • Figure 4: Graphs of $\log(\epsilon_\lambda))$ and $\log(1-\chi(\lambda)))$ as functions of $\mu=\lambda^2$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Theorem 3.6
  • ...and 18 more