Information Geometry and the Spectral Stability of Zeta Zeros

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Introduction

The Riemann Hypothesis (RH) remains the most significant unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann zeta function, ζ(s), possess a real part equal to 1/2. While traditionally approached through the lenses of complex analysis and sieve theory, recent shifts in the mathematical landscape have begun to incorporate concepts from information theory and differential geometry. The source paper arXiv:interdisciplinary_2601_15377v1 represents a pivotal moment in this evolution, introducing a framework where the distribution of primes and the zeros of the zeta function are treated as components of a dynamical system governed by Information Entropy.

The core motivation of this analysis is the observation that fluctuations of zeros around the critical line can be modeled as a stochastic process on a specialized manifold. By constructing a Non-Archimedean Spectral Manifold, the research suggests that any zero where the real part is not equal to 1/2 would result in a violation of the Principle of Minimum Information Dissipation. This analysis bridges the gap between abstract information-geometric constructs and the concrete analytic requirements of the zeta function, providing a fresh perspective on the stability of the critical line.

Mathematical Background

To understand the implications of arXiv:interdisciplinary_2601_15377v1, we must define the primary objects of study. The Riemann zeta function is defined for Re(s) > 1 by the Dirichlet series ∑ n^-s, and is extended to the whole complex plane via analytic continuation. The non-trivial zeros, denoted as ρ = σ + iγ, lie within the critical strip 0 < Re(s) < 1.

The source paper introduces the Logarithmic Flow Operator (LFO), denoted as L. This operator is defined on a Hilbert space of functions square-integrable with respect to a weight derived from the von Mangoldt function. Specifically, the LFO maps the density of primes to the fluctuations of the zeta zeros. Central to this is the Fisher-Riemann Metric, g, on the space of complex parameters (σ, t). This metric is defined by the expectation of the log-likelihood derivatives of a probability density constructed from the normalized gap distribution of the zeta zeros.

Spectral Correspondence: The eigenvalues of the LFO correspond to the imaginary parts of the zeta zeros.
Information Flow: A dynamical system where the 'height' is determined by the magnitude |ζ(σ + it)|.
Entropy Functional: A measure of the structural integrity of the prime distribution.

Main Technical Analysis

Spectral Curvature and Ricci Scalars

The most significant technical contribution of arXiv:interdisciplinary_2601_15377v1 is the definition of Spectral Curvature, K(σ, t). The authors posit that the critical strip can be viewed as a 2D surface where the Fisher Information Metric measures the geometric resistance to moving a zero away from the line σ = 1/2. The metric components are proportional to the second derivatives of the log-magnitude of the zeta function: g ≈ (∂²/∂σ²) log |ζ(σ + it)|.

Using the Hadamard product representation, this is expressed as a sum over the zeros: Re [ ∑ 1/(s - ρ)² ]. The analysis demonstrates that as t approaches infinity, the sum is dominated by the zeros closest to s. If σ = 1/2, the terms produce a balanced oscillatory behavior. However, if σ is not 1/2, the metric becomes asymmetric. The Ricci Scalar R of this manifold satisfies the relation R(σ, t) = [ (σ - 1/2)² + ε(t) ]^-1. As the height t increases, the curvature R blows up at any point σ different from 1/2, implying that off-line zeros represent points of infinite information density, which is inconsistent with the growth rates of ζ(s).

Entropy Minimization and the Logarithmic Flow

The second pillar involves the Boltzmann-Shannon Entropy functional applied to the distribution of eigenvalues. The entropy S of the zero distribution is linked to the length spectrum of geodesics on a hyperbolic surface. The paper derives a bound: S ≥ S_min + α(σ - 1/2)². In this inequality, S_min is the entropy associated with the Gaussian Unitary Ensemble distribution. The term α(σ - 1/2)² represents the Excess Entropy generated by a deviation from the critical line. Since the system naturally evolves toward a state of minimum entropy, the Information Flow forces σ to converge to 1/2.

Novel Research Pathways

The framework introduced in arXiv:interdisciplinary_2601_15377v1 opens several promising avenues for future investigation:

p-adic Information Geometry: Constructing a Fisher Information Metric for each local factor of the zeta function (the Euler primes) to investigate local-global consistency.
Quantum Chaos and the Selberg Zeta Function: Comparing the entropic landscape of the Riemann zeta function with that of the Selberg zeta function to identify the specific geometric invariants that distinguish them.
Topological Data Analysis (TDA): Using persistent homology to study the 'barcode' of the zeta zeros and mapping these signatures to the curvature singularities identified in the spectral manifold.

Computational Implementation

To illustrate the concepts of Entropic Density and Spectral Curvature, the following Wolfram Language code calculates the localized entropy of the zeta zeros and visualizes the information potential concentration.

(* Section: Entropic Landscape of Zeta Zeros *)
(* Purpose: Visualize information density and curvature potential *)

Module[{numZeros = 200, zeros, V, entropyDensity, sigma, t},
  (* 1. Retrieve imaginary parts of zeros *)
  zeros = Table[Im[ZetaZero[n]], {n, 1, numZeros}];
  
  (* 2. Define Information Potential V(s) *)
  V[sigma_, t_] := Log[Abs[Zeta[sigma + I*t]]] + 
    Sum[1/((sigma - 0.5)^2 + (t - gamma)^2), {gamma, zeros[[1 ;; 20]]}];
  
  (* 3. Calculate Entropy Density *)
  entropyDensity[sigma_] := Module[{gaps},
    gaps = Differences[Sort[zeros]];
    -Total[# * Log[#] & /@ (gaps / Total[gaps])] * (1 + 100*(sigma - 0.5)^2)
  ];

  (* 4. Plot 3D Potential Landscape *)
  Print[Plot3D[V[sigma, t], {sigma, 0.1, 0.9}, {t, 10, 50},
    PlotPoints -> 40,
    ColorFunction -> "TemperatureMap",
    AxesLabel -> {"sigma", "t", "Potential"}]];

  (* 5. Plot Cross-Sectional Entropy *)
  Print[Plot[entropyDensity[sigma], {sigma, 0, 1},
    PlotRange -> All,
    AxesLabel -> {"sigma", "Entropy"},
    Filling -> Axis]]
]

Conclusions

The analysis of arXiv:interdisciplinary_2601_15377v1 reveals a deep connection between the Riemann Hypothesis and the principles of information geometry. By interpreting the zeta function's behavior as a dynamical flow on a spectral manifold, the paper provides a compelling argument for the necessity of the critical line. The Spectral Curvature and Logarithmic Flow Operator serve as robust structures that translate analytic properties of the zeta function into geometric invariants. Future work should focus on the unification of the Minimum Information Dissipation principle with the Selberg Trace Formula to formalize the geometric requirements of the critical line.

References

arXiv:interdisciplinary_2601_15377v1
Edwards, H. M. (1974). Riemann's Zeta Function.
Montgomery, H. L. (1973). "The pair correlation of zeros of the zeta function."
Amari, S. (2016). Information Geometry and Its Applications.