Asymptotic Determinants and the Spectral Positivity of the Riemann Xi Function

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Executive Summary

The research paper arXiv:cel-01215340_file_ZetaPositivityTalk proposes a sophisticated "positivity route" to the Riemann Hypothesis (RH) by examining the Riemann xi-function in a normalized form. By analyzing the function g(u), an entire function evaluating the Xi function along imaginary arguments, the study establishes a bridge between the distribution of zeta zeros and the asymptotic behavior of logarithmic derivatives. The key insight lies in the coefficient matrices a_ij(-u), which exhibit specific sign patterns and growth rates. While a specific 2x2 minor is shown to be asymptotically negative due to the dominance of Gamma-factor curvature, this result serves as a diagnostic tool for refining positivity criteria. This approach is promising as it transforms the global horizontal distribution problem of the Riemann Hypothesis into a local analytic investigation of series coefficients and spectral measures.

Introduction

The Riemann Hypothesis remains the most significant unsolved problem in analytic number theory, asserting that all non-trivial zeros of the Riemann zeta function zeta(s) lie on the critical line Re(s) = 1/2. A standard strategy to address this involves the Riemann xi-function, xi(s), and its related form Xi(z) = xi(1/2 + iz). In this formulation, RH is equivalent to the statement that all zeros of Xi(z) are real. This "real zeros" requirement naturally interfaces with the theory of the Laguerre-Polya class and total positivity.

The source paper arXiv:cel-01215340_file_ZetaPositivityTalk contributes to this discourse by investigating the "positivity" of the zeta function through the lens of infinite product representations and moment-generating series. The analysis focuses on a specific transformation g(u), which allows for the derivation of explicit asymptotic expansions for the coefficients of its logarithmic derivative. By understanding how these coefficients behave as u approaches infinity, researchers can identify the structural constraints that the zeta function must satisfy to maintain its zeros on the critical line.

Mathematical Background

The foundation of this analysis rests on the Riemann Xi function, defined as Xi(z) = xi(1/2 + iz). This function is even and entire, and its zeros correspond to the non-trivial zeros of the zeta function. The source paper introduces a normalized function g(u) defined as:

g(u) = Xi(i * sqrt(u)) / Xi(0)

Substituting the standard definition of the xi-function, we obtain the explicit representation:

g(u) = [(1 - 4u) / (pi * sqrt(u) / 2)] * [Gamma(1/4 + sqrt(u)/2) * zeta(1/2 + sqrt(u))] / [Gamma(1/4) * zeta(1/2)]

Under the assumption of the Riemann Hypothesis, Xi(z) admits a Hadamard product over its zeros gamma_n, which leads to the linear factor representation in the u-plane: product(1 - lambda_n z), where lambda_n = 1/gamma_n². For RH to hold, the sequence lambda_n must be positive, summable, and bounded below 1. The paper further relates this to a moment expansion involving a positive kernel phi(t):

product(1 - lambda_n z) = sum [(-z)ⁿ / (2n)!] * [integral(t²ⁿ * phi(t) dt) / integral(phi(t) dt)]

This identity connects the discrete zero data to the continuous moments of phi(t), providing a dual framework for testing zero locations.

Main Technical Analysis

Asymptotic Decay and the Determinant Obstruction

The core technical contribution of arXiv:cel-01215340_file_ZetaPositivityTalk involves the derivation of asymptotic expansions for the logarithmic derivatives of g(u). These are packaged into coefficients a_ij(-u), which are defined as follows:

a₀₀(-u): Proportional to the negative second derivative of log g(u).
a₀₁(-u): Proportional to the third derivative of log g(u).
a₁₁(-u): Proportional to the negative fourth derivative of log g(u).

For large values of u, the zeta function zeta(1/2 + sqrt(u)) approaches 1 rapidly, meaning the derivatives are dominated by the Gamma function and the pre-factors. The source provides the following explicit expansions:

a₀₀(-u) = (log u - log(4 * pi²) - 2) / (16 * u^3/2) + 7 / (8 * u²) + O(u^-5/2)

a₀₁(-u) = (3 * log u - log(64 * pi³) - 8) / (64 * u^5/2) + 7 / (8 * u³) + O(u^-7/2)

A critical finding is the behavior of the 2x2 determinant D(u) = a₀₀(-u)a₁₁(-u) - a₀₁(-u)². The expansion reveals:

D(u) = - (3 * log u - log(64 * pi⁶) - 8)² / (4096 * u⁵) - 7 * (3 * log u - log(64 * pi⁶) - 8) / (256 * u^11/2) + O(u^-6)

The leading term is strictly negative for large u. This indicates that the total positivity of the xi-function coefficients fails at high scales. This is not a contradiction of RH, but rather an indication that the Archimedean component (the Gamma factor) imposes a curvature that overwhelms the arithmetic signal of the zeros at infinity.

Spectral Properties and Zero Distribution

The negativity of the determinant suggests that g(u) does not belong to the strongest possible class of totally positive functions. However, the lambda_n sequence remains positive under RH. The analysis demonstrates that the log u terms in the expansion originate from the Stirling approximation of the Gamma function. This implies that the "positivity" of the zeta function is robust; it is a structural property of the completed zeta function that is maintained as long as the zeros remain on the critical line.

Novel Research Pathways

1. Renormalized Spectral Density

Since the Gamma factor dominates the asymptotic behavior and causes the determinant negativity, a promising research direction is the renormalization of g(u). By dividing out the Archimedean factor (the Gamma and power components), one can isolate the arithmetic part of the xi-function. Analyzing the positivity of the derivatives of this renormalized function would provide a more sensitive diagnostic for the location of the zeros, free from the overwhelming curvature of the Gamma function.

2. Higher-Order Positivity Tensors

The current study focuses on the 2x2 matrix of coefficients. Expanding this to n x n Hankel matrices of logarithmic derivatives could reveal a deeper hierarchy of sign patterns. If RH is true, these determinants must satisfy specific conditions related to the Turan inequalities. Investigating the "first failure index"—the order at which a determinant changes sign—could provide a numerical bound on the height of zeros on the critical line.

3. Operator-Theoretic Realization

The identity product(1 - lambda_n z) = sum (moments) suggests that the lambda_n values can be viewed as eigenvalues of a compact operator. A novel pathway involves constructing a Hilbert space and a self-adjoint operator whose spectrum matches the lambda_n sequence. If such an operator can be constructed using the kernel phi(t), the positivity of lambda_n (and thus RH) would follow from the operator's spectral properties.

Computational Implementation

Wolfram Language

(* Section: Asymptotic Determinant Analysis *)
(* Purpose: Compute derivatives of log g(u) and verify the determinant sign *)

Module[{
  uVal = 1000, 
  prec = 50, 
  g, logG, a00, a01, a11, det
},
  (* Define the normalized Xi function g(u) *)
  g[u_] := ((1 - 4 u)/(Pi Sqrt[u]/2)) * 
    (Gamma[1/4 + Sqrt[u]/2] Zeta[1/2 + Sqrt[u]]) / 
    (Gamma[1/4] Zeta[1/2]);

  (* Define Logarithmic derivatives *)
  logG[u_] := Log[g[u]];
  
  (* a00 is -d^2/du^2, a01 is 1/2 d^3/du^3, a11 is -1/6 d^4/du^4 *)
  a00 = -D[logG[u], {u, 2}] /. u -> uVal;
  a01 = (1/2) * D[logG[u], {u, 3}] /. u -> uVal;
  a11 = -(1/6) * D[logG[u], {u, 4}] /. u -> uVal;
  
  det = a00 * a11 - a01^2;
  
  Print["Results at u = ", uVal];
  Print["a00: ", N[a00, 10]];
  Print["a01: ", N[a01, 10]];
  Print["a11: ", N[a11, 10]];
  Print["Determinant (a00*a11 - a01^2): ", N[det, 10]];

  (* Plotting the determinant over a range to see the sign *)
  Plot[a00 * a11 - a01^2, {u, 500, 2000}, 
    PlotLabel -> "Determinant a00*a11 - a01^2", 
    AxesLabel -> {"u", "Det"},
    PlotStyle -> Red]
]

Conclusions

The analysis of the Riemann Xi function through the positivity of its transformed coefficients, as detailed in arXiv:cel-01215340_file_ZetaPositivityTalk, offers a compelling bridge between analytic number theory and the theory of entire functions. While the asymptotic negativity of the determinant a₀₀a₁₁ - a₀₁² reveals an obstruction to simple total positivity, it also clarifies the dominant role of the Archimedean factor in the xi-function's structure. The most promising next steps involve renormalizing these expansions to isolate the arithmetic signal and pursuing the operator-theoretic spectral realizations of the zero-based sequence lambda_n.

References

Source Paper: arXiv:cel-01215340_file_ZetaPositivityTalk
Li, X.-J. (1997). The Positivity of a Sequence of Numbers and the Riemann Hypothesis. Journal of Number Theory, 65(2).
Bombieri, E., and Lagarias, J. C. (1999). Complements to Li's Criterion for the Riemann Hypothesis. Journal of Number Theory, 77(2).