Order Statistics and Meta-Analytic Aggregation: A Spectral Pathway to the Riemann Hypothesis

Introduction

The search for a resolution to the Riemann Hypothesis (RH) has transitioned from pure arithmetic into a spectral and probabilistic endeavor. Since the observation that the spacings of the non-trivial zeros of the Riemann zeta function ζ(s) mirror the eigenvalues of the Gaussian Unitary Ensemble (GUE), the spectral approach has become a dominant paradigm. This article proposes a novel entry point by drawing from the mathematical framework of meta-analytic signal detection, specifically the Rank of Optimal P-values (rOP) method described in arXiv:1407.8376v1.

In quantitative systems, the meta-analysis problem involves aggregating weak signals from K independent sources to identify a significant global effect. The rOP method provides a robust statistic for detecting signals present in at least a subset of studies, even when the majority report noise. We contend that the Riemann zeta function can be modeled as a meta-analytic aggregator of prime-based detectors, where the zeros on the critical line (Re(s) = 1/2) represent optimal consensus points of an infinite collection of oscillators.

Mathematical Structures in Aggregation Theory

The core of the rOP framework involves the vector of p-values P = (p₁, p₂, ..., p_K). Under the null hypothesis H₀, each p_k is independently and uniformly distributed on the interval [0, 1]. The rOP statistic is defined by the r-th order statistic:

S_r = p_(r)

where p₍₁₎ ≤ p₍₂₎ ≤ ... ≤ p_(K) are the ordered p-values. Under H₀, S_r follows a Beta distribution with parameters r and K-r+1. The probability density function (PDF) is defined as:

f(x; r, K) = [K! / ((r-1)!(K-r)!)] * x^r-1 * (1-x)^K-r

Beyond order statistics, the framework utilizes linear aggregators such as Fisher's method, T_F = -2 * Σ log(p_k), which follows a Chi-squared distribution with 2K degrees of freedom, and Stouffer's method, T_S = Σ Φ^-1(1 - p_k), where Φ^-1 is the inverse normal CDF. These operators map a K-dimensional hypercube of evidence onto a 1-dimensional decision manifold, a process we define as Information Aggregation.

Analogies to Analytic Number Theory

Zero Spacings as Calibrated P-Values

Let the non-trivial zeros be 1/2 + iγ_n. The normalized spacings are defined as s_n = (γ_n+1 - γ_n) * (log(γ_n/2π) / 2π). According to the Montgomery-Odlyzko law, these spacings follow the GUE distribution. If F is the CDF for the spacing distribution, then the transformation p_n = 1 - F(s_n) produces pseudo p-values that are approximately Uniform(0,1). The rOP statistic then provides a robust measure for detecting anomalies in zero distributions across different scales.

The Prime Sieve as a Meta-Analysis

The Sieve of Eratosthenes can be viewed as an intersection hypothesis test (HS_A). Let each prime p_k be a study testing the primality of an integer n. The p-value of a prime p_k relative to n is p(n, p_k) = (n mod p_k) / p_k. An integer is detected as prime if it yields a significant result across all studies p_k < n^1/2. The Riemann Hypothesis, via the error term in the Prime Number Theorem, asserts that the noise in this meta-analysis is minimized according to a square-root law, analogous to optimal False Discovery Rate (FDR) control.

Novel Research Pathways

Pathway 1: rOP-Robust Explicit Formula Inequalities

We propose a research program to replace single-kernel explicit formula arguments with replicability across kernels. Let {g_k} be a family of test functions in Weil's explicit formula. Each yields a statistic T_k = Σ G_k(ρ). Under RH, these T_k must satisfy specific positivity constraints. By calibrating T_k into p-values, we can use the rOP aggregator to detect off-line zeros that would force coherent anomalies across multiple kernel views, gaining robustness against local dependence.

Pathway 2: Stouffer-Type Aggregation of Mollifiers

Mollifiers M(s) = Σ a_n n^-s are used to approximate 1/ζ(s). We propose constructing a family of mollifiers M_k and using Stouffer's method to find an optimal linear combination. This leads to a Generalized Least Squares problem in the Hilbert space of Dirichlet polynomials, where the weights are chosen to minimize the variance of the aggregated probe, potentially sharpening zero-density estimates.

Pathway 3: FDR Control for Li Coefficients

The Li criterion states that RH is equivalent to λ_n > 0 for all n. We propose an adaptive thresholding scheme, similar to the Benjamini-Hochberg procedure, to allocate computational resources for certifying these inequalities. By treating each λ_n as a hypothesis and bounding the approximation error e_n, one could establish a convergent error budget Σ P(failure) < ∞, transforming finite computation into a logically complete proof for the infinite sequence.

Computational Investigation

The following Wolfram Language code tests the analogy between the rOP Beta law and the distribution of zeta zero spacings using Wigner-surmise calibration.

(* Title: rOP Beta-law from calibrated zeta zero spacings *)
Module[{n0 = 100, n1 = 1100, gam, gaps, dens, s, a, Fg, pvals, K = 10, r = 3, blocks, rop, betaCDF, empCDF, xs},
  gam = N[Table[Im[ZetaZero[n]], {n, n0, n1}]];
  gaps = Differences[gam];
  dens = (Log[Most[gam]/(2 Pi)])/(2 Pi);
  s = gaps * dens;
  a = 4/Pi;
  Fg[x_] := 1 - Exp[-a x^2] * (1 + a x^2);
  pvals = 1 - Fg /@ s;
  blocks = Partition[pvals, K, K, 1, {}];
  rop = Sort[#][[r]] & # /@ blocks;
  betaCDF[x_] := CDF[BetaDistribution[r, K - r + 1], x];
  xs = Range[0.01, 0.99, 0.01];
  empCDF = Table[Mean[Boole[rop <= x]], {x, xs}];
  ListLinePlot[{Transpose[{xs, empCDF}], Transpose[{xs, betaCDF /@ xs}]},
    PlotStyle -> {Blue, {Red, Dashed}},
    PlotLegends -> {"Empirical rOP", "Theoretical Beta"},
    AxesLabel -> {"x", "P(S_r <= x)"}]
]

Conclusions

The mathematical structures of the rOP method provide a rigorous toolkit for aggregating information across the heterogeneous "probes" of the Riemann zeta function. The most promising pathway involves treating the critical line as a Minimax FDR Manifold, where the information density of the prime distribution is optimally conserved. Future work should focus on constructing a Meta-Zeta operator whose spectral density is governed by these order-statistic aggregation laws.

References

• Source Paper: arXiv:1407.8376v1
• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function.
• Benjamini, Y., and Hochberg, Y. (1995). Controlling the false discovery rate.
• Odlyzko, A. M. (1987). On the distribution of spacings between zeros of the zeta function.