Asymptotic Positivity and Spectral Obstructions in the Riemann Xi Function

Executive Summary

The research presented in arXiv:cel-01215340v1_document_document investigates a family of positivity constraints associated with the Riemann xi-function, Ξ(z). By constructing a normalized ratio, g(u), which relates the values of the function along the imaginary axis to its value at the origin, the analysis identifies a sequence of differential invariants, a_ij(-u). The paper provides sharp large-u asymptotic expansions for these coefficients, revealing that a critical 2x2 determinant, a₀₀(-u)a₁₁(-u) - a₀₁(-u)², becomes asymptotically negative.

This result provides a rigorous obstruction to naive global positivity conjectures that attempt to prove the Riemann Hypothesis (RH) through simple Hankel-type inequalities. However, the analysis also demonstrates that this negativity is primarily an archimedean artifact driven by the Gamma factor. This approach is promising because it distinguishes between the deterministic background of the xi-function and the arithmetic fluctuations of the zeta zeros, offering a clear path toward renormalization strategies that could isolate the true RH-sensitive signals.

Introduction

The Riemann Hypothesis remains the central challenge of analytic number theory, asserting that the non-trivial zeros of the Riemann zeta function, ζ(s), lie strictly on the critical line where the real part of s is 1/2. A common strategy in modern research involves translating this geometric requirement into the language of positivity. If the zeros are real, then associated functions must belong to the Laguerre-Pólya class, which implies a vast array of inequalities involving higher-order derivatives.

The source paper arXiv:cel-01215340v1_document_document sits at the intersection of this positivity tradition and asymptotic analysis. It focuses on the modified function Ξ(z) = ξ(1/2 + iz), which is real-valued for real z. The core problem addressed is the behavior of Ξ on the imaginary axis, which corresponds to the growth of ζ(s) for large real s. By analyzing the logarithmic derivatives of a normalized version of this function, the author extracts coefficients that encode the spectral data of the zeta zeros.

This analysis contributes a vital cautionary result: while many positivity conditions are necessary for RH, they can be swamped by the growth of the archimedean factor (the Gamma function). By providing explicit asymptotic expansions for the matrix coefficients of the xi-function, the paper allows researchers to see through the "archimedean noise" and identify where the arithmetic truth of the Riemann Hypothesis actually resides.

Mathematical Background

The primary object of study is the completed zeta function, ξ(s), defined as (1/2)s(s-1)π^-s/2Γ(s/2)ζ(s). The function Ξ(z) is then defined as ξ(1/2 + iz). Under the Riemann Hypothesis, all zeros of Ξ(z) are real. If we denote these zeros as ±γ_n, we can write the Hadamard product:

Ξ(z) / Ξ(0) = ∏_n=1^∞ (1 - z²/γ_n²)

The source paper introduces a transformation to a variable u by setting z = i multiplied by the square root of u. This defines the auxiliary function:

g(u) = Ξ(i sqrt(u)) / Ξ(0) = (1 - 4u) / (π sqrt(u) / 2) * [Γ(1/4 + sqrt(u)/2) ζ(1/2 + sqrt(u))] / [Γ(1/4) ζ(1/2)]

The coefficients a_ij(-u) are derived from the derivatives of the logarithm of g(u). Specifically, a₀₀ corresponds to the negative second derivative, a₀₁ to the third derivative, and a₁₁ to the fourth derivative, with appropriate factorial scaling. These coefficients are linked to the moments of a positive kernel φ(t), which represents the Fourier transform of the xi-function. The paper assumes standard convergence properties for the sequence λ_n = 1/γ_n², namely that the sum of λ_n is finite and the supremum of the sequence is less than 1.

Main Technical Analysis

Spectral Properties and Asymptotic Expansions

The paper arXiv:cel-01215340v1_document_document performs a rigorous asymptotic evaluation of the coefficients as u tends to infinity. Using Stirling's formula for the Gamma function and the fact that ζ(s) approaches 1 for large real s, the following expansions are established:

• a₀₀(-u): (log u - log(4π²) - 2) / (16 u^3/2) + 7 / (8 u²) + 1 / (64 u^5/2) + 9 / (16 u³) + O(u^-7/2)
• a₀₁(-u): (3 log u - log(64π⁶) - 8) / (64 u^5/2) + 7 / (8 u³) + O(u^-7/2)
• a₁₁(-u): O(u^-9/2)

The hierarchy of decay rates (u^-3/2, u^-5/2, and u^-9/2) indicates that the higher-order derivatives of the xi-function are increasingly dominated by the algebraic and logarithmic terms of the archimedean factor. This structure is a direct consequence of the functional equation and the order of the xi-function.

The Determinantal Obstruction

A major finding of the analysis is the behavior of the determinant D(u) = a₀₀(-u)a₁₁(-u) - a₀₁(-u)². The paper demonstrates that for large u:

D(u) = - (3 log u - log(64π⁶) - 8)² / (4096 u⁵) - 7 (3 log u - log(64π⁶) - 8) / (256 u^11/2) + O(u^-6)

The leading term is negative for all sufficiently large u. This is a significant result because it shows that a common intuition—that these derivative-based matrices should be positive definite—is false in the global limit. This negativity does not disprove the Riemann Hypothesis; rather, it identifies that the "positivity" required by RH is more subtle than the raw derivatives of the xi-function can provide.

Archimedean vs. Arithmetic Contributions

The logarithmic terms in the expansions (log u) arise from the Gamma function and the pre-factors of the xi-function. In contrast, the contribution from the zeta zeros (the arithmetic part) is exponentially small along the real axis, behaving like O(2^-sqrt(u)). This explains why the large-u behavior is dominated by the negative determinant. The arithmetic contribution, which contains the information about whether zeros are on the critical line, is "swamped" by the archimedean terms in this specific differential representation.

Novel Research Pathways

1. Archimedean Renormalization

The most immediate pathway is the construction of renormalized coefficients, A_ij(u), where the known archimedean contribution is subtracted. By defining A_ij(u) = a_ij(u) - a_ij,arch(u), researchers can isolate the arithmetic remainder. If the Riemann Hypothesis holds, these renormalized coefficients should exhibit a different positivity signature, one that is not destroyed by the logarithmic growth of the Gamma factor.

2. Stieltjes Transform of the Reciprocal Function

A second direction involves shifting the focus from g(u) to its reciprocal, h(u) = 1/g(u). Under RH, g(u) is a product of terms (1 + λ_n u) with positive λ_n. This implies that h(u) should be a completely monotone function. Testing whether (-1)^k h^(k)(u) is non-negative for all k and u offers a more robust positivity test that is naturally aligned with the product structure of the xi-function.

3. Oscillatory Correction Analysis

While the leading terms are smooth, the actual xi-function contains oscillatory corrections due to the distribution of the zeros. A promising pathway involves using high-precision numerical methods to extract these oscillations from the residuals of the asymptotic expansions. The frequency and phase of these residuals should map directly to the imaginary parts of the zeta zeros, providing a new way to verify the critical line property through asymptotic consistency.

Computational Implementation

This Wolfram Language implementation computes the normalized function g(u) and the determinant D(u) to demonstrate the asymptotic sign transition and the influence of the zeta zeros.

(* Section: Analysis of Xi-Function Derivative Determinants *)
(* Purpose: Demonstrate the large-u negativity of the determinant D(u) *)

Module[{
  (* Define g(u) using the xi function relation *)
  g, a00, a01, a11, detD, uVal
},
  g[u_] := RiemannXi[I Sqrt[u]] / RiemannXi[0];

  (* Define coefficients as logarithmic derivatives *)
  (* We use high precision to handle the growth of the Gamma factor *)
  a00[u_] := -D[Log[g[x]], {x, 2}] /. x -> u;
  a01[u_] := (1/2) * D[Log[g[x]], {x, 3}] /. x -> u;
  a11[u_] := -(1/6) * D[Log[g[x]], {x, 4}] /. x -> u;

  detD[u_] := a00[u] * a11[u] - (a01[u])^2;

  (* Generate values and check for the sign transition *)
  Print["Evaluating Determinant D(u) at various scales:"];
  Print[TableForm[
    Table[{uVal, N[detD[uVal], 20]}, {uVal, {10, 50, 100, 500, 1000}}],
    TableHeadings -> {None, {"u", "D(u) Value"}}
  ]];

  (* Plot the function to visualize the asymptotic behavior *)
  Plot[detD[u], {u, 100, 1000}, 
    PlotRange -> All, 
    AxesLabel -> {"u", "D(u)"}, 
    PlotLabel -> "Asymptotic Negativity of the a_ij Determinant",
    GridLines -> Automatic]
]

Conclusions

The analysis of arXiv:cel-01215340v1_document_document provides a critical refinement of the positivity program for the Riemann Hypothesis. By deriving the explicit large-u asymptotics of the xi-function's logarithmic derivatives, the paper demonstrates that global positivity is obstructed by the archimedean factor. This finding shifts the research focus toward renormalization—subtracting the deterministic Gamma-driven background to reveal the arithmetic signals of the zeta zeros. The most promising next steps involve applying these asymptotic filters to higher-order derivative chains and investigating the complete monotonicity of the reciprocal xi-function as a more stable criterion for the critical line.

References

• arXiv:cel-01215340v1_document_document
• Edwards, H. M. (1974). Riemann's Zeta Function. Academic Press.
• Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function. Oxford University Press.