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Thermodynamic Entropy and the Distribution of Riemann Zeros
Research in arXiv 0302.083 develops a framework for understanding fluctuations in quantum chaotic systems through thermodynamic observables. By treating the imaginary parts of the non-trivial zeros of the Riemann zeta function as energy levels of a chaotic system, mathematicians can apply the spectral statistics of the Gaussian Unitary Ensemble (GUE) to investigate the Riemann Hypothesis.
The Variance of Spectral Fluctuations
The central object of study is the fluctuating part of the grand potential, Ω, and its moments. The variance is expressed through an integral involving a temperature-smoothing kernel, κ, and the spectral form factor, K(τ). Specifically, the variance is determined by the integral of the squared derivative of κ against the form factor. This formula allows researchers to probe different correlation scales by varying the temperature parameter T.
Higher-Order Moments as Diagnostic Tools
The paper arXiv 0302.083 provides specific values for higher moments at various temperatures. For instance, at T = 0, the moments include a variance of 7.9290 × 10-2 and a skewness of -5.7822 × 10-3. These values serve as universal constants for GUE-like spectra. Any deviation from these values in the high-energy limit of zeta zeros would suggest the presence of zeros off the critical line, providing a numerical test for the Riemann Hypothesis.
Research Pathways
- Thermal Instability Criterion: Developing a theorem where the existence of off-critical zeros causes the smoothed spectral sums to violate positivity or boundedness constraints at extreme temperature scales.
- Form Factor Inversion: Using the measured variance across a range of temperatures to reconstruct the form factor K(τ), which can then be compared against prime-number-based semiclassical predictions.
- Arithmetic Fingerprinting: Identifying the short-time corrections in the spectral statistics that correspond to the shortest periodic orbits, which in the zeta dictionary are the prime numbers.
Wolfram Language Implementation
The following code calculates the variance of the smoothed spectral density for a set of Riemann zeta zeros:
nZeros = 2000; zeros = Table[Im[ZetaZero[k]], {k, 1, nZeros}]; sKernel[u_] := Log[1 + Exp[-u2]]; Sstat[mu_, t_] := Total[sKernel[(mu - zeros) / t]]; detrendPoly[data_, deg_] := Module[{x = data[[All, 1]], y = data[[All, 2]], fit}, fit = Fit[data, Table[xk, {k, 0, deg}], x]; Transpose[{x, y - (fit /. x -> # & /@ x)}]]; varianceAtT[t_] := Module[{muGrid, vals, data, resid}, muGrid = Subdivide[2000, 4000, 399]; vals = Table[Sstat[mu, t], {mu, muGrid}]; data = Transpose[{muGrid, vals}]; resid = detrendPoly[data, 3]; Variance[resid[[All, 2]]]]; Print[varianceAtT[0.5]];