Stochastic Stability and Information Geometry of the Riemann Critical Line

Introduction

The Riemann Hypothesis (RH) remains a profound unsolved problem in mathematics, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. While traditional approaches rely on analytic number theory, the source paper arXiv:interdisciplinary_2601_15769v1, titled "Information-Theoretic Stability of the Riemann Critical Line via Stochastic Quantization," introduces a perspective from statistical mechanics and information theory.

The core motivation of arXiv:interdisciplinary_2601_15769v1 is to treat the distribution of prime numbers as a dynamical system under stochastic fluctuations. By mapping the behavior of the zeta function onto a manifold of information states, the research suggests that the critical line acts as a point of maximum stability or a ground state. This provides a physical intuition: any deviation from the line σ = 1/2 corresponds to a state of higher entropy disallowed by the underlying symmetries of the prime-generating process.

Mathematical Background

The Riemann zeta function ζ(s) is defined for Re(s) > 1 by the series ∑ n^-s. Through analytic continuation, it is extended to the complex plane. The functional equation relates ζ(s) to ζ(1-s), implying symmetry around the critical line. The source paper arXiv:interdisciplinary_2601_15769v1 introduces the Stochastic Zeta Operator (L_ζ), which acts on a Hilbert space of functions over the critical strip.

This operator is constructed using the log-derivative of the zeta function, where the von Mangoldt function Λ(n) is treated as a set of impulses in a continuous field. A key innovation is the definition of the Information Metric g_ij on the critical strip, where the distance between points is determined by the covariance of fluctuations in the Dirichlet series. The paper connects these structures to the Li Criterion, suggesting that the positivity of Li constants is a consequence of the operator's positive-definiteness.

Main Technical Analysis

The Zeta-Langevin Equation

The authors of arXiv:interdisciplinary_2601_15769v1 propose that the value of the zeta function along a vertical line can be modeled by a Langevin-type equation: dζ/dt = -∇V(ζ) + η(t). In this model, V(ζ) represents a potential well derived from prime distribution, and η(t) is a Gaussian white noise term accounting for pseudo-random prime gaps. The paper argues that for σ = 1/2, the potential V reaches a global minimum, effectively trapping zeros on the critical line.

Information Geometric Curvature

The analysis calculates the Fisher Information Curvature of the zeta function's value distribution. By analyzing the Hessian of the log-likelihood, the researchers demonstrate that the curvature diverges as s approaches the critical line. This divergence suggests a phase transition. In this framework, the Riemann Hypothesis is re-interpreted as a requirement for Topological Stability: the critical line is the only manifold where information flow is perfectly balanced.

Moment Estimates and Growth Rates

The paper provides new bounds on the 2k-th moments of the zeta function using stochastic quantization. These estimates align with the Keating-Snaith conjecture based on Random Matrix Theory (RMT). The distribution of zeros follows Gaussian Unitary Ensemble (GUE) statistics because the Langevin dynamics are unitary in the limit of infinite height T. By linking the entropy production rate to zero density, the paper supports the standard Riemann-von Mangoldt formula with a vanishingly small error term.

Novel Research Pathways

• Pathway 1: Non-Equilibrium Thermodynamics: Extend the Langevin model into a thermodynamic treatment where a "Zeta-Entropy" function S(σ) is minimized at σ = 1/2. This would relate the work required to move a zero away from the critical line to free energy differences.
• Pathway 2: Quantum Graph Models: Develop quantum graph analogs where eigenvalue distributions of the Laplacian on arithmetic graphs approximate zeta zero statistics. This offers finite-dimensional approximations for numerical investigation of spectral gaps.

Computational Implementation

(* Section: Spectral Analysis of Riemann Zeta Zeros *)
(* Purpose: Visualize the distribution of zero spacings and compare with GUE *)

Module[{numZeros, zeros, spacings, avgSpacing, normalizedSpacings, gueDist, spacingPlot, theoreticalPlot, entropy},
  numZeros = 100;
  zeros = Table[Im[ZetaZero[n]], {n, 1, numZeros}];

  (* Calculate normalized spacings *)
  spacings = Differences[zeros];
  avgSpacing = Mean[spacings];
  normalizedSpacings = spacings / avgSpacing;

  (* Define GUE spacing distribution *)
  gueDist[s_] := (32/Pi^2) * s^2 * Exp[-(4/Pi) * s^2];

  (* Create Histogram *)
  spacingPlot = Histogram[normalizedSpacings, {0.2}, "PDF", 
    PlotLabel -> "Normalized Spacings of Zeta Zeros",
    AxesLabel -> {"s (Normalized Spacing)", "Probability Density"}];

  (* Overlay GUE curve *)
  theoreticalPlot = Plot[gueDist[s], {s, 0, 3}, PlotStyle -> {Red, Thick}];

  (* Calculate Information Entropy *)
  entropy = -Total[# * Log[#] & /@ (BinCounts[normalizedSpacings, {0, 3, 0.1}] / numZeros)];
  Print["Estimated Information Entropy of Spacings: ", entropy];

  Show[spacingPlot, theoreticalPlot, PlotRange -> All]
]

Conclusions

The analysis in arXiv:interdisciplinary_2601_15769v1 provides a compelling lens by reframing zero distribution as a problem of stochastic stability. The derivation of the Zeta-Langevin equation suggests that the Riemann Hypothesis is a fundamental property of information flow in complex systems. Future research should focus on unifying Information Geometric Curvature with the Li Criterion to establish if curvature divergence necessitates the alignment of zeros on the critical line.

References

• arXiv:interdisciplinary_2601_15769v1
• Keating, J. P., & Snaith, N. C. (2000). Random Matrix Theory and ζ(1/2 + it).
• Li, X.-J. (1997). The Positivity of a Sequence of Numbers and the Riemann Hypothesis.