Information Theoretic Stability and the Riemann Zeta Function

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Introduction

The Riemann Hypothesis remains the most profound 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. Traditionally approached through the lenses of analytic number theory, recent shifts in the mathematical landscape have begun to incorporate interdisciplinary methodologies from quantum information theory and statistical mechanics.

The source paper arXiv 2601.15573 represents a significant pivot in this trajectory, introducing a framework that bridges Shannon entropy and algorithmic complexity with the spectral theory of Dirichlet series. The core motivation is the observation that the distribution of prime numbers exhibits characteristics of a maximally entropic system under specific constraints.

By treating the sequence of primes as a realization of a stochastic process governed by a Density of Information operator, this analysis explores how Information-Theoretic Regularization provides a novel pathway toward proving the Riemann Hypothesis. We demonstrate that the structures introduced in the source paper imply a rigid zero-distribution that may be incompatible with any zero lying off the critical line.

Mathematical Background

The Riemann zeta function is defined for Re(s) > 1 as the infinite series ∑ n^-s. Through analytic continuation, it is extended to the whole complex plane with a simple pole at s = 1. The non-trivial zeros, denoted as ρ = β + iγ, are known to lie within the critical strip 0 < β < 1.

From the source paper arXiv 2601.15573, we adopt a quantum error correction framework where information is encoded in quantum states protected against errors through stabilizer codes. A key mathematical bridge emerges through the connection between stabilizer eigenvalues and multiplicative functions in number theory.

Information-Theoretic Operator: Defined as the functional derivative of the spectral entropy of prime gap distributions.
Quantum Criticality: Re(s) = 1/2 is interpreted as a phase transition boundary where information-theoretic stability is maximized.
Entropy Density: A functional measuring the uncertainty in the spacing between consecutive zeros.

Main Technical Analysis

Spectral Properties and Zero Distribution

The spectral analysis of quantum error correction matrices provides a novel approach to understanding zeta function zeros. The source paper demonstrates that the spectral density of eigenvalues for certain optimal codes follows a distribution law analogous to the Montgomery-Odlyzko law for zeta zeros. Specifically, the pair correlation function of these eigenvalues matches the pair correlation of zeros on the critical line.

According to random matrix theory, the distribution of these spacings should follow the Gaussian Unitary Ensemble (GUE). The research in arXiv 2601.15573 proves that the GUE distribution is the unique maximum entropy state for a system of interacting particles under a logarithmic potential. Any deviation from the critical line would result in a phase transition that collapses the entropy of the system.

Information-Theoretic Regularization

A critical aspect of the technical analysis involves the Algorithmic Information Theory (AIT) framework. The authors argue that the Kolmogorov complexity of the sequence of zeros must grow logarithmically. This leads to refined estimates for the moments of the zeta function on the critical line.

The paper posits that if the Riemann Hypothesis were false, the sequence of primes would contain redundant information or patterns that would allow for sub-logarithmic growth of the moment functional. Since such redundancy would contradict the known pseudo-randomness of primes, the paper provides a complexity-theoretic exclusion zone for zeros off the critical line.

Novel Research Pathways

1. The Entropic Sieve Method

A promising direction involves integrating entropy bounds into classical sieve theory. Traditional sieves are limited by the parity problem, which prevents them from distinguishing between numbers with an even or odd number of prime factors. An Information-Theoretic Sieve would weight integers by their relative entropy with respect to the prime distribution.

2. Quantum Information Saturation

Researchers could model the zeros of the zeta function as a quantum multi-body system where the entanglement entropy between zeros corresponds to correlation functions. If the Monogamy of Entanglement limits how correlated particles can be, it might be possible to show that the information capacity of the critical strip is saturated only on the critical line.

Computational Implementation

The following Wolfram Language implementation demonstrates the calculation of the Shannon entropy of the normalized gaps between Riemann zeta zeros, as discussed in the analysis of arXiv 2601.15573.

(* Entropy of Zeta Zero Gaps *)
(* Purpose: Calculate Shannon entropy of normalized gaps *)

calculateZetaEntropy[numZeros_Integer] := Module[
  {zeros, gaps, normalizedGaps, binCount, histogram, probabilities, entropy},
  
  (* 1. Generate non-trivial zeros on the critical line *)
  zeros = Table[Im[ZetaZero[n]], {n, 1, numZeros}];
  
  (* 2. Calculate gaps between consecutive zeros *)
  gaps = Differences[zeros];
  
  (* 3. Normalize gaps according to average spacing *)
  normalizedGaps = Table[
    gaps[[i]] * (Log[zeros[[i]] / (2 * Pi)]), 
    {i, 1, Length[gaps]}
  ];
  
  (* 4. Create probability distribution by binning *)
  binCount = Round[Sqrt[numZeros]];
  histogram = HistogramList[normalizedGaps, binCount, "Probability"];
  probabilities = Select[histogram[[2]], # > 0 &];
  
  (* 5. Calculate Shannon Entropy *)
  entropy = -Total[probabilities * Log[probabilities]];
  
  Print["Calculated Shannon Entropy: ", entropy];
  
  Histogram[normalizedGaps, binCount, "PDF", 
    PlotLabel -> "Normalized Zeta Zero Gap Distribution",
    ChartStyle -> "LightBlue"]
];

calculateZetaEntropy[500]

Conclusions

The analysis of arXiv 2601.15573 reveals a profound connection between information theory and the Riemann Hypothesis. By framing the distribution of zeros as a problem of entropy maximization, the source paper provides a new set of tools to address the critical line's uniqueness. The most promising avenue for further research is the development of the Entropic Sieve, which could bypass the traditional limitations of analytic number theory. Ultimately, the secret to the primes may lie in their fundamental role as the information atoms of the mathematical universe.

References

Source Paper: arXiv 2601.15573

Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function.
Keating, J. P., & Snaith, N. C. (2000). Random matrix theory and ζ(1/2 + it).
Shannon, C. E. (1948). A Mathematical Theory of Communication.