If our chaotic operator is derived correctly, then the Riemann hypothesis holds true

TL;DR

This paper constructs an operator-theoretic framework anchored in the Riemann–von Mangoldt formula to explore the nontrivial zeros of the Riemann zeta function through dynamics, spectral theory, and random matrix models. It introduces a density-driven discrete map on the critical line, analyzes its chaotic features via Lyapunov exponents, and reveals its limitations in reproducing fine-scale zero statistics. Building on this, it defines a self-adjoint chaotic operator $oxed{\mathcal{O}_eta}$ on a weighted Hilbert space with weight $w(T)=rac{dN}{dT}$, proves essential self-adjointness and unboundedness, and studies its spectral resolution. Finite-dimensional truncations yield Hermitian matrices whose eigenvalue statistics align with GUE predictions and Odlyzko data, and exhibit a hydrogen-atom–like spectral shape, supporting the view that $oxed{oxed{\mathcal{O}_eta}}$ belongs to the same universality class as the nontrivial zeta zeros, in line with Hilbert–Pólya-type programs. While not proving RH, the work provides a concrete, testable framework bridging arithmetic, chaotic dynamics, and random matrix theory with potential for sharper spectral results and deeper insights into zeta zeros.

Abstract

This work develops an operator-theoretic and dynamical framework inspired by the Riemann--von Mangoldt formula, chaotic dynamics, and random-matrix models for the Riemann zeta function, without attempting to prove the Riemann Hypothesis. Starting from the explicit zero-counting function $N(T)$, we construct a discrete map on the critical line and analyse its Lyapunov exponents and bifurcation diagrams, showing that the smooth von Mangoldt term generates a strongly unstable flow that captures the global growth of the zero density. Motivated by this dynamics, we define a self-adjoint ``chaotic'' operator $\mathcal{O}_α$ on a weighted Hilbert space with weight $\mathrm{d}N/\mathrm{d}T$, prove its unboundedness and essential self-adjointness, and describe its spectral resolution via the spectral theorem. Finite-dimensional truncations of $\mathcal{O}_α$ yield Hermitian random matrices whose eigenvalue statistics agree numerically with Gaussian unitary ensemble predictions and show qualitative similarities to both Odlyzko's zeta zeros and the hydrogen-atom spectrum, suggesting that $\mathcal{O}_α$ lies in the same universality class as the nontrivial zeros and providing a concrete Hilbert--Pólya--type framework rather than a proof of the conjecture.

Figures (6)

• Figure 1: Left: largest Lyapunov exponent $\lambda_1$ as a function of $\Delta T$ for the map $T_{n+1} = T_n + \Delta T\,dN/dT$. The exponents are strictly positive and increase with $\Delta T$, reflecting exponential sensitivity to initial conditions. Right: comparison between the first $30$ Odlyzko zeros (blue dots) and the trajectory of the density-based map with $\Delta T = 1.17$ (red dashed), started at the first zero. The map follows the global trend but rapidly overshoots, illustrating that the smooth density alone cannot reproduce the actual zero sequence.
• Figure 2: Left: empirical gap distribution of the first $50$ Riemann zeros, with mean $\mu_{\text{zeros}} \approx 2.72$. Right: gap distribution for the trajectory of the density-based map with $\Delta T = 1.17$, with mean $\mu_{\text{map}} \approx 20.12$ and a pronounced heavy tail. The contrast between the two histograms highlights that the deterministic density flow captures only the coarse growth of $N(T)$ and fails to reproduce the local level statistics.
• Figure 3: Finite–time Lyapunov exponent as a function of the iteration step for an orbit started at the first nontrivial zero with a fixed step size $\Delta T$. The convergence to a positive value reveals exponential sensitivity to initial conditions in the density-based map.
• Figure 4: Bifurcation diagram of the density-based map for $\Delta T \in [0.01,1.0]$, obtained by plotting the last $800$ iterates after discarding an initial transient. The fan–like structure for small $\Delta T$ and the saturation near $T \approx 400$ reflect the strong instability of the coarse-grained dynamics generated by the Riemann–von Mangoldt density.
• Figure 5: Histogram of eigenvalues for a truncated random matrix associated with the chaotic operator. The clustering in a narrow band and the absence of degeneracies are consistent with GUE-type level repulsion.

Theorems & Definitions (16)

• Lemma 1: Chaos Threshold
• Proof 1: Sketch
• Definition 1: Weighted Hilbert space and generator
• Lemma 2: Symmetry of $H_0$
• Proof 2
• Definition 2: Chaotic operator
• Theorem 1: Essential self-adjointness
• Proof 3: Sketch
• Corollary 1: Spectral theorem and diagonalization
• Conjecture 1: Hilbert--Pólya realization via $\mathcal{O}_\alpha$