Discrete Keiper-Li Coefficients and the Spectral Distribution of Riemann Zeros

Executive Summary

The research paper arXiv:cea-01274195_file_text introduces a significant advancement in the study of the Riemann Hypothesis (RH) by developing a discretized deformation of the classical Keiper-Li coefficients. Traditionally, Li's criterion establishes that RH is equivalent to the positivity of a sequence of coefficients derived from high-order derivatives of the Riemann xi-function at s=1. These derivatives are numerically unstable and analytically complex to compute. The source paper bypasses this by constructing a sequence Λ_n expressed as a finite sum over the values of the xi-function at even integers. By utilizing the known relations between ξ(2m), Bernoulli numbers, and the double factorial, the author provides a framework that is both explicitly computable and rigorously sensitive to the distribution of zeros. This approach transforms the analytic complexity of the Riemann zeta function into a problem of combinatorial asymptotics and spectral filtering, offering a new pathway for verifying the RH through the zeros' staircase counting function N(T).

Introduction

The Riemann Hypothesis remains the most profound unsolved problem in number theory, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. In the 1990s, the work of Keiper and Li introduced real sequences whose global positivity or growth rates serve as equivalent conditions for the RH. However, the computational bottleneck of these sequences lies in their recursive nature and the extreme precision required for numerical differentiation. The paper arXiv:cea-01274195_file_text addresses these limitations by discretizing the derivatives into finite differences anchored at even integers 2, 4, ..., 2n. This transformation leverages the functional equation and the explicit values of ξ(2m), allowing for the direct calculation of any individual coefficient Λ_n. This article analyzes the mathematical backbone of this discretization, its sensitivity to off-critical zeros, and the novel research directions it opens for spectral analysis of the zeta function.

Mathematical Background

The analysis relies on the completed Riemann xi-function, defined as ξ(s) = (1/2) s(s-1) π^(-s/2) Γ(s/2) ζ(s), which satisfies the functional equation ξ(s) = ξ(1-s). The source paper utilizes special values of this function at even integers:

• Special Values: 2ξ(2m) = |B_2m| / |(2m-3)!!| * (2π)^m, where B_2m are Bernoulli numbers.
• Double Factorial: The definition is extended for all odd integers k such that k!! = 2^((k+1)/2) Γ(k/2 + 1) / √π. This ensures that (-1)!! = 1.
• Weight Coefficients: The sequence Λ_n is constructed using weights A_nm defined as: A_nm = 2^(-2n) / (2m-1) * Binomial(2(n+m), n+m) * Binomial(n+m, 2m).

In terms of Gamma functions, these weights are expressed as A_nm = [2^(2m) Γ(n+m+1/2)] / [(2m-1) (n-m)! (2m)! √π]. These structures allow the discretization of the logarithmic derivative of the xi-function into a finite, alternating sum.

Main Technical Analysis

Discretized Sequence and RH Sensitivity

The core innovation is the definition of the sequence Λ_n = (-1)^n ∑ (-1)^m A_nm log 2ξ(2m), where the sum runs from m=1 to n. Unlike the original Li coefficients, which are tied to a single basepoint (s=1), Λ_n samples the xi-function across a range of values. The paper proves that this sequence retains "RH sensitivity":

• If RH is True: The sequence exhibits tame growth, specifically Λ_n ≈ (1/2) log n + c, where c is a constant related to the Euler-Mascheroni constant and log 2π.
• If RH is False: Any zero ρ with Re(ρ) = σ > 1/2 introduces a polynomial growth term proportional to n^(σ-1/2). This creates a clear numerical signature that would eventually dominate the logarithmic term.

Integral Representation and Zero Distribution

The paper establishes a profound connection between Λ_n and the zeros' staircase counting function N(T) through a sine-integral transform. The coefficients can be represented as the integral of 2 sin Θ_n(θ) N( (1/2) cot(θ/2) ) over the interval [0, π]. This formula demonstrates that the behavior of Λ_n is a direct reflection of the "evenness" of the distribution of zeros on the critical line. The rational kernel used in the discretization acts as a spectral filter, magnifying deviations from the expected density of zeros.

Numerical Precision and Cancellation

A critical observation in arXiv:cea-01274195_file_text is the requirement for high precision. Since A_nm involves large binomial coefficients and the sum is alternating, significant cancellation occurs. The minimum required precision peaks at approximately log10 of the maximum term in the sum. However, because the values of 2ξ(2m) are known exactly in terms of π and Bernoulli numbers, this method is fundamentally more stable than numerical differentiation of the zeta function at high orders.

Novel Research Pathways

Pathway 1: Saddle-Point Asymptotics for Bernoulli Sums

A promising theoretical direction involves using the saddle-point method to derive a rigorous asymptotic expansion for the A_nm-weighted sums of log|B_2m|. By identifying the stationary phase of the alternating sum, researchers could potentially prove the (1/2) log n growth rate unconditionally or establish explicit error bounds that define the threshold for detecting off-critical zeros.

Pathway 2: Arithmetic Prime-Power Expansion

The source paper suggests an arithmetic form for Λ_n by substituting the Euler product of the zeta function into the sum. This would express the coefficients as a sum over prime powers p^r. Investigating the weights assigned to each prime power could reveal whether the positivity of the sequence is a direct consequence of the Prime Number Theorem's error term, effectively bridging prime density with the geometry of the critical line.

Pathway 3: Generalized Basepoint Deformation

The current discretization uses even integers. A novel research path would involve varying the anchor points s_m of the finite difference scheme (e.g., using a quadratic or exponential spacing). This could allow for "localized" sensitivity, where the growth of the sequence detects zeros within specific vertical strips of the critical region, potentially improving the efficiency of numerical RH tests.

Computational Implementation

The following Wolfram Language code computes the discretized Λ_n coefficients and compares them to the theoretical logarithmic growth rate predicted under the Riemann Hypothesis.

(* Section: Discretized Keiper-Li Sequence *) 
(* Purpose: Compute Lambda_n using Bernoulli numbers and check growth *)

ClearAll[Anm, LambdaN, XiValue];

(* Define the weights A_nm *)
Anm[n_, m_] := (2^(-2 n) / (2 m - 1)) * Binomial[2 (n + m), n + m] * Binomial[n + m, 2 m];

(* Define 2*Xi(2m) using Bernoulli numbers *)
XiValue[m_] := Module[{doubleFact},
  doubleFact = 2^((2 m - 3 + 1)/2) * Gamma[(2 m - 3)/2 + 1] / Sqrt[Pi];
  Abs[BernoulliB[2 m]] / Abs[doubleFact] * (2 * Pi)^m
];

(* Compute Lambda_n sequence *)
ComputeLambda[nMax_, prec_] := Table[
  Module[{sum},
    sum = N[Sum[(-1)^m * Anm[n, m] * Log[XiValue[m]], {m, 1, n}], prec];
    (-1)^n * sum
  ], 
  {n, 1, nMax}
];

(* Compare with theoretical growth (1/2) log n + c *)
constantC = 0.5 * (EulerGamma - Log[2 * Pi] - 1);
nMax = 30;
results = ComputeLambda[nMax, 100];
theoretical = Table[0.5 * Log[n] + constantC, {n, 1, nMax}];

(* Output results for n=10, 20, 30 *)
TableForm[Table[{n, results[[n]], theoretical[[n]]}, {n, {10, 20, 30}}], 
  TableHeadings -> {None, {"n", "Lambda_n", "Theoretical"}}]

(* Plot the comparison *)
ListLinePlot[{results, theoretical}, 
  PlotLegends -> {"Discretized Lambda_n", "(1/2) log n + c"}, 
  AxesLabel -> {"n", "Value"}, 
  PlotLabel -> "Growth of Discretized Coefficients"]

Conclusions

The discretization of Keiper-Li coefficients presented in arXiv:cea-01274195_file_text provides a powerful bridge between the analytic properties of the Riemann zeta function and discrete combinatorial sums. By utilizing special values at even integers, the author has created a sequence Λ_n that is both numerically accessible and theoretically deep. The most promising avenue for further research lies in the refinement of the integral representation involving the zeros' staircase function, which could lead to new spectral criteria for the Riemann Hypothesis. Ultimately, this approach suggests that the alignment of zeros on the critical line is encoded in the delicate cancellation of Bernoulli-weighted sums, a realization that may guide future attempts at a formal proof.

References

• arXiv:cea-01274195_file_text: Discretized Keiper-Li coefficients and the Riemann Hypothesis.
• Keiper, J. B. (1992). Power series expansions of Riemann's xi function. Mathematics of Computation, 58(198).
• Li, X.-J. (1997). The positivity of a sequence of numbers and the Riemann hypothesis. Journal of Number Theory, 65(2).