Spectral Stability and Information Geometry: A New Frontier for the Riemann Hypothesis

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, denoted as ζ(s), possess a real part equal to 1/2. Traditional approaches have focused on complex analysis and analytic number theory, but recent developments in arXiv:interdisciplinary_2601_15340v1 introduce a paradigm shift. This research situates the distribution of these zeros within the context of non-equilibrium statistical mechanics, spectral operator theory, and information geometry.

The motivation for this interdisciplinary study stems from the Hilbert-Pólya conjecture, which suggests that the imaginary parts of the zeros of ζ(s) correspond to the eigenvalues of a self-adjoint operator. The source paper arXiv:interdisciplinary_2601_15340v1 extends this intuition by constructing a dynamical system where the flow of prime numbers corresponds to a geodesic on a manifold of probability distributions. This article provides a technical decomposition of how spectral bounds and information metrics might necessitate the confinement of zeros to the critical line.

Mathematical Background

The Riemann zeta function is defined for Re(s) > 1 by the absolutely convergent series ζ(s) = Σ n^-s. The functional equation relates ζ(s) to ζ(1-s), implying a symmetry about the critical line Re(s) = 1/2. The spectral framework developed in arXiv:interdisciplinary_2601_15340v1 centers on a family of differential operators L_λ acting on L²(R⁺) spaces. These operators take the form L_λf(x) = -f''(x) + V_λ(x)f(x), where the potential function V_λ(x) incorporates weighted prime counting functions through the construction of delta functions at the logarithms of primes.

The spectrum of these operators consists of discrete eigenvalues whose distribution exhibits statistical correlations with the imaginary parts of zeta zeros. If ρ = 1/2 + iγ denotes a non-trivial zero, the quantities γ_n share spacing statistics with the eigenvalues under appropriate normalization. This connection is formalized through a spectral zeta function that maps operator data to arithmetic data, providing a bridge between functional analysis and the distribution of primes.

Spectral Properties and Information Stability

The core technical contribution involves the trace formula relating eigenvalue sums to zeta function values. For the operator family defined, we establish the relationship where the asymptotic expansion of the heat trace encodes information about the moments of the Riemann zeta function on the critical line. The leading term provides the Weyl asymptotic formula for the number of eigenvalues, which corresponds to the known asymptotic for zeta zeros. This correspondence extends to higher-order terms, where the heat trace expansion encodes the error term in the zero counting function.

Furthermore, arXiv:interdisciplinary_2601_15340v1 introduces an Information Metric Tensor g_ij defined over the space of Dirichlet series. By treating the zeros as singularities of information flow in a high-dimensional manifold, the paper demonstrates that the fluctuations in arithmetic entropy are minimized when the zeros are perfectly aligned on the critical line. This connects the Montgomery-Odlyzko law to the stability of the transfer operator's spectrum, suggesting that any "off-line" zero would create an energetically unfavorable arithmetic vacuum that violates the second law of thermodynamics as applied to arithmetic systems.

Novel Research Pathways

• Spectral Rigidity and Zero Correlation: Investigating quantitative connections between spectral rigidity measures for the operator family and correlation functions for zeta zeros. Precise agreement between the rigidity growth and zeta zero statistics would provide strong supporting evidence for the hypothesis.
• Arithmetic Information Metric on Selberg Class: Extending the information metric to the entire Selberg Class of L-functions. Researchers can investigate whether the information curvature can detect the presence of Landau-Siegel zeros or other anomalies in the critical strip.
• Computational Spectral Verification: Developing high-precision algorithms to compute eigenvalues of the operator family L_λ. This allows for testing statistical predictions about zeta zero distribution at scales beyond current direct zeta computation limits.

Computational Implementation

(* Section: Spectral Analysis of Zeta-Related Operators *)
(* Purpose: Compute eigenvalues of operators encoding prime information *)

ConstructPrimePotential[xmax_, npts_] :=
  Module[{dx, xvals, potential, primes, logprimes, pos},
   dx = xmax/npts;
   xvals = Range[0, xmax - dx, dx];
   potential = ConstantArray[0., npts];
   primes = Prime[Range[PrimePi[Exp[xmax]]]];
   logprimes = Log[primes];
   Do[pos = Ceiling[logprimes[[i]]/dx];
    If[pos <= npts, potential[[pos]] += Log[primes[[i]]]/dx],
    {i, Length[primes]}];
   {xvals, potential}];

ConstructSpectralMatrix[potential_, dx_] :=
  Module[{n, matrix, i},
   n = Length[potential];
   matrix = SparseArray[{}, {n, n}];
   Do[If[i > 1, matrix[[i, i - 1]] = -1/dx^2];
    matrix[[i, i]] = 2/dx^2 + potential[[i]];
    If[i < n, matrix[[i, i + 1]] = -1/dx^2];, {i, n}];
   matrix];

AnalyzeSpectrum[xmax_, npts_, neigs_] :=
  Module[{potential, dx, matrix, eigenvals, xvals},
   {xvals, potential} = ConstructPrimePotential[xmax, npts];
   dx = xmax/npts;
   matrix = ConstructSpectralMatrix[potential, dx];
   eigenvals = Sort[Re[Eigenvalues[matrix, -neigs, Method -> "Arnoldi"]]];
   eigenvals];

(* Example Usage *)
eigenvalues = AnalyzeSpectrum[10.0, 1000, 50];
Print["First 10 eigenvalues:", eigenvalues[[1 ;; 10]]];

Conclusions

This investigation into the spectral and geometric properties of the Riemann zeta function, as proposed in arXiv:interdisciplinary_2601_15340v1, provides a robust framework for understanding the critical line distribution. By integrating operator theory and information geometry, the research suggests that the Riemann Hypothesis is not merely a statement about roots, but a requirement for the stability of arithmetic information. The most promising avenue for immediate progress lies in the computational verification of spectral rigidity, which could provide independent confirmation of zeta zero behavior at unprecedented scales.

References

Source Paper: arXiv:interdisciplinary_2601_15340v1

Montgomery, H.L. (1973). "The pair correlation of zeros of the zeta function." Proceedings of Symposia in Pure Mathematics.

Keating, J.P. & Snaith, N.C. (2000). "Random matrix theory and ζ(1/2+it)." Communications in Mathematical Physics.