Information-Theoretic Stability and the Stochastic Quantization of Zeta Zeros

Introduction

The Riemann Hypothesis (RH) remains the most profound unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. While traditionally approached through analytic number theory, the source paper arXiv:interdisciplinary_2601_14062v1 introduces a paradigm shift by applying concepts from information theory and non-gaussian stochastic dynamics to the study of the zeta function. This article provides a technical analysis of these findings, connecting them to the Hilbert-Polya conjecture and spectral statistics.

The core motivation of the source material is the observation that the distribution of prime numbers mimics the behavior of complex physical systems at the edge of chaos. By treating the critical strip as a phase space for a specific class of stochastic processes, the authors attempt to demonstrate that any deviation from the critical line would result in a violation of information-theoretic stability. This work bridges the gap between the analytic properties of zeta and the statistical properties of the Gaussian Unitary Ensemble (GUE).

Mathematical Background

The Riemann zeta function is defined for Re(s) > 1 by the Dirichlet series sum of n^-s from n=1 to infinity. Through analytic continuation, it is extended to the entire complex plane with a simple pole at s = 1. The functional equation relates the value of the function at s to its value at 1-s, establishing a symmetry around the critical line.

The paper arXiv:interdisciplinary_2601_14062v1 introduces a q-deformed zeta landscape using a Spectral Density Function, which represents the density of zeros along the imaginary axis. A central object in this framework is the Stochastic Operator, whose stability is linked to the location of the zeros. The authors define the Arithmetic Entropy of a set of zeros based on the probability density of their normalized spacings. The hypothesis is that for the system to remain in arithmetic equilibrium, entropy must be maximized, a state that occurs only if the real part of the zeros is exactly 1/2.

Main Technical Analysis

Spectral Properties and Zero Distribution

The spectral analysis framework provides tools for investigating the distribution of zeros through the lens of random matrix theory. The connection between the statistical properties of eigenvalue spacings in random Hermitian matrices and the spacings between consecutive zeta zeros is fundamental. The pair correlation function exhibits the same statistical behavior as eigenvalue correlations in the GUE.

A crucial innovation in arXiv:interdisciplinary_2601_14062v1 lies in its treatment of spectral rigidity and number variance. These quantities measure the deviation of eigenvalue statistics from Poissonian behavior. The number variance for intervals of length L shows logarithmic growth, and the source paper provides refined error estimates that suggest this behavior is a requirement for the stability of the underlying arithmetic manifold.

Stochastic Quantization and the Critical Line

The primary technical innovation is the application of stochastic quantization. The authors treat the real and imaginary parts of the zeta function as coordinates in a two-dimensional Langevin equation. They demonstrate that the ground state of this dynamical system corresponds precisely to the zeros on the critical line. If a zero were to exist off the line, the resulting force in the Langevin dynamics would create a non-zero flux in the information manifold, leading to a divergence in the Fisher Information Metric.

Sieve Bounds and Prime Density

The paper also provides insights into the connections between the Riemann Hypothesis and sieve theory. It shows that the distribution of prime numbers can be estimated using specific sieve bounds that are closely related to the distribution of the zeros. By minimizing the Tsallis entropy for the zero distribution, the authors arrive at an optimal growth bound for the zeta function, which is consistent with the Lindelof Hypothesis.

Novel Research Pathways

The q-Generalized Explicit Formula

The classical explicit formula connects primes to the zeros of the zeta function. We propose developing a q-deformed Explicit Formula based on the non-extensive entropy framework. This would involve replacing the standard Von Mangoldt function with a q-weighted version to investigate if the sum over zeros converges more rapidly for specific values of q.

Information Geometry of the Zeta Landscape

Using the Fisher Information Metric defined in arXiv:interdisciplinary_2601_14062v1, one could study the Ricci flow of the zeta information manifold. The Riemann Hypothesis can be reformulated as a statement that the critical line is a geodesic of minimal entropy. This pathway could prove that the critical line is an attractor for the zeros under renormalization group flow.

Topological Data Analysis (TDA) of Zero Spacings

We suggest using Persistent Homology to analyze the shape of zero distributions. By constructing complexes from sequences of normalized zero spacings, researchers can identify higher-order topological invariants that remain invariant under the functional equation, potentially revealing spectral gaps predicted by the stochastic operator.

Computational Implementation

The following Wolfram Language implementation demonstrates the information-theoretic analysis of zeta zeros. It calculates the normalized spacings and computes the Shannon Entropy to test the Spectral Rigidity hypothesis.

(* Section: Information Entropy of Zeta Zero Spacings *)
(* Purpose: Evaluate the entropy of the distribution of non-trivial zeros *)

(* 1. Generate the first 1000 imaginary parts of the non-trivial zeros *)
zeros = Table[Im[ZetaZero[n]], {n, 1, 1000}];

(* 2. Calculate the normalized spacings *)
normalizedSpacings = Table[
  (zeros[[n + 1]] - zeros[[n]]) * (Log[zeros[[n]]] / (2 * Pi)),
  {n, 1, Length[zeros] - 1}
];

(* 3. Define a binning function for the probability distribution *)
binWidth = 0.1;
bins = BinCounts[normalizedSpacings, {0, 4, binWidth}];
probabilities = bins / Total[bins];

(* 4. Calculate Shannon Entropy *)
shannonEntropy = -Total[Map[If[# > 0, # * Log[2, #], 0] &, probabilities]];

(* 5. Compare with GUE theoretical entropy *)
guePDF[s_] := (32/Pi^2) * s^2 * Exp[-4 * s^2 / Pi];
theoreticalEntropy = -NIntegrate[guePDF[s] * Log[2, guePDF[s]], {s, 0, 10}];

(* 6. Output Results *)
Print["Calculated Shannon Entropy: ", shannonEntropy];
Print["Theoretical GUE Entropy: ", theoreticalEntropy];

Histogram[normalizedSpacings, {binWidth}, "Probability", 
 PlotLabel -> "Distribution of Normalized Zeta Zero Spacings"]

Conclusions

The analysis of arXiv:interdisciplinary_2601_14062v1 reveals a profound connection between the Riemann Hypothesis and the principles of information-theoretic stability. By framing the distribution of zeros as a consequence of stochastic quantization, the paper provides a novel physical justification for the critical line. The most promising avenue for progress lies in the spectral analysis direction, where computational frameworks can investigate higher-order correlations and test deviations from universality that encode arithmetic information.

References

• arXiv:interdisciplinary_2601_14062v1
• Montgomery, H. L. (1973). "The pair correlation of zeros of the zeta function."
• Odlyzko, A. M. (1987). "On the distribution of spacings of zeros of the zeta function."