Mapping the Critical Line: A New Look at the Keiper-Li Coefficients and the Riemann Hypothesis

Executive Summary

The research paper arXiv:cea-01274195v1_document presents a profound analytical refinement of the Keiper-Li criteria for the Riemann Hypothesis (RH). By investigating the sequence of coefficients λ_n derived from the logarithmic derivatives of the Riemann Xi function ξ(s), the author establishes a rigorous framework for evaluating the distribution of the non-trivial zeros of the Riemann Zeta function. The key technical insight lies in the construction of a specific combinatorial kernel, denoted as A_nm, which facilitates a more stable numerical and analytical approach to Li's positivity criterion. The paper demonstrates that the positivity of these coefficients is not merely a binary indicator of RH but is deeply linked to the staircase counting function of the zeros, N(T). This approach is promising because it transforms a global analytic problem into a sequence-based convergence problem, allowing for the use of asymptotic analysis and discrete transform methods. By bridging the gap between the discrete sum over zeros and the continuous integral representations of the ξ function, the work provides a pathway to quantify the closeness of the zeros to the critical line through the growth rates of λ_n.

Introduction

The Riemann Hypothesis remains the most significant unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann Zeta function ζ(s) possess a real part equal to 1/2. In the late 20th century, Keiper (1992) and Li (1997) introduced a sequence of real numbers, λ_n, whose behavior is inextricably linked to the location of these zeros. Specifically, Li's criterion states that RH is true if and only if λ_n is greater than zero for every positive integer n. The computational challenge lies in efficiently calculating these coefficients for large n while maintaining numerical precision.

The source paper, arXiv:cea-01274195v1_document, addresses the inherent numerical instabilities and theoretical complexities of the original Li coefficients. The specific problem addressed is the slow convergence and high precision required to evaluate the sums involving the ξ function at large n. The author's contribution is the introduction of a modified sequence, Λ_n, and a detailed analysis of the combinatorial weights A_nm. This analysis allows for a more granular view of how individual zeros ρ contribute to the aggregate sum. By employing conformal mappings that pull the critical line to the boundary of the unit disk, the paper translates the analytic properties of ξ(s) into the power series coefficients of a transformed function.

Mathematical Background

To understand the derivations in the source paper, one must first define the Riemann Xi function: ξ(s) = (1/2) s (s-1) π^-s/2 Γ(s/2) ζ(s). This function is entire and satisfies the functional equation ξ(s) = ξ(1-s). The non-trivial zeros of ζ(s) are precisely the zeros of ξ(s). The Keiper-Li coefficients are defined via the expansion of log(2ξ(s/(s-1))) as a power series.

The source paper introduces the structure A_nm, a combinatorial weighting factor: A_nm = 2^-2n / (2m-1) * Binomial(2(n+m), n+m) * Binomial(n+m, 2m). Equivalently, this can be expressed using the double factorial (k!!), where k!! = k(k-2)...1 for odd k > 0. For odd k < 0, the paper utilizes the Gamma function extension: k!! = 2^(k+1)/2 Γ(k/2 + 1) / sqrt(π). Notably, this definition yields (-1)!! = 1.

The paper also connects the ξ function at even integers to the Bernoulli numbers B_2m: 2ξ(2m) = |B_2m| / |(2m-3)!!| * (2π)^m. This relationship is critical because it allows the coefficients to be expressed in terms of rational numbers and powers of π, facilitating high-precision computation and theoretical bounding.

Main Technical Analysis

Conformal Mappings and the Unit Disk

A central theme of the analysis is the use of the conformal mapping M(z) = 1 / (1-z). This mapping pulls the critical line Re(s) = 1/2 to the unit circle |z| = 1. Under this transformation, the Riemann Hypothesis is equivalent to the requirement that the function f(z) = log(2ξ(M(z))) be regular (analytic) within the entire open unit disk |z| < 1. The integral representation of the modified coefficients Λ_n involves a contour integral reminiscent of a Blaschke product, where specific terms serve to isolate the contributions of specific points in the complex plane.

The Asymptotic Growth and Residue Calculus

The paper provides a rigorous derivation of the asymptotic behavior of λ_n. Under the assumption of the Riemann Hypothesis, the coefficients follow the growth law λ_n = n (1/2 log n + c) + o(n), where c = 1/2 (γ - log 2π - 1) which is approximately -1.1303. If RH were false, and there existed a zero with Re(ρ) = σ > 1/2, the coefficients would grow exponentially. The paper refines this by showing that Λ_n can be decomposed into a sum over zeros F_n(ρ), allowing one to detect off-line zeros by looking for oscillations or exponential growth that deviates from the expected trend.

The Staircase Function and Integral Representations

One of the most technically demanding sections involves the integral: λ_n = ∫ 2 sin(nθ) N(1/2 cot(θ/2)) dθ, where N(T) is the zero-counting function. This links the discrete sequence to the continuous fluctuations of the zeros. The term 2 sin(nθ) acts as a high-frequency filter that extracts information from the staircase function. The paper argues that only a finer analysis of the limiting integral can definitively prove the positivity for all n, as the error terms become significant for large indices.

Novel Research Pathways

1. Stability Analysis of the Positivity Kernel: A promising direction is to investigate the stability of the positivity of λ_n under small perturbations of the zeros. Given the integral representation, one could model the zeros as a point process, such as the GUE ensemble, and determine the probability that the coefficients remain positive. This would connect the deterministic Li criterion to the probabilistic framework of Random Matrix Theory.

2. Optimized Kernels for Off-line Zero Detection: Research could focus on designing a family of kernels K_n(s) whose discrete residue evaluation still uses special values but whose zero-residue response has larger magnitude for Re(ρ) > 1/2. This would lower the detectability threshold for hypothetical off-line zeros at moderate values of n.

3. Arithmetic of the u_n Sequence: The paper introduces a sequence u_n that isolates the arithmetic components of the ξ function. Investigating the p-adic properties of this sequence or its connection to the arithmetic of cyclotomic fields could provide a non-analytic proof of positivity, independent of complex analysis.

Computational Implementation

The following Wolfram Language code calculates the modified Keiper-Li coefficients and compares them to the theoretical asymptotic growth predicted in the paper.

(* Section: Keiper-Li Coefficient Computation *)
(* Purpose: To demonstrate the calculation of coefficients from special values *)

Module[{maxN = 30, xi, logXiExp, coefficients, nLogN, c},
  
  (* Define the Riemann Xi function *) 
  xi[s_] := 1/2 s (s - 1) Pi^(-s/2) Gamma[s/2] Zeta[s];
  
  (* Expand Log[2 xi[s]] via conformal mapping z = (s-1)/s *) 
  logXiExp = Series[Log[2 xi[1/(1 - z)]], {z, 0, maxN}];
  
  (* Extract coefficients: lambda_n = n * [z^n] logXiExp *) 
  coefficients = Table[n * SeriesCoefficient[logXiExp, n], {n, 1, maxN}];
  
  (* Theoretical asymptotic: n * (1/2 Log[n] + c) *) 
  c = 1/2 (EulerGamma - Log[2 Pi] - 1);
  nLogN = Table[n * (1/2 Log[n] + c), {n, 1, maxN}];
  
  (* Print results for the first 10 terms *) 
  Print["n | Computed Lambda_n | Asymptotic Value"];
  Do[Print[n, " | ", N[coefficients[[n]], 5], " | ", N[nLogN[[n]], 5]], {n, 1, 10}];
  
  (* Plot comparison *) 
  ListLinePlot[{coefficients, nLogN}, 
    PlotLegends -> {"Computed Lambda_n", "Asymptotic"}, 
    AxesLabel -> {"n", "Value"}, 
    PlotLabel -> "Growth of Keiper-Li Coefficients"]
]

Conclusions

The investigation into arXiv:cea-01274195v1_document reveals that the Riemann Hypothesis is encoded in the fine-grained balance of combinatorial weights and the logarithmic values of the Xi function. The transition from the original Li coefficients to the modified sequence Λ_n provides a more stable analytical foundation, reducing the impact of numerical noise and highlighting the underlying structural properties of the zeros. The most promising avenue for further research is the integration of the integral representation with the statistical distribution of zeros. If the oscillations in N(T) can be bounded using the explicit formula of prime numbers, it may be possible to show that the staircase never deviates far enough from its smooth approximation to allow the coefficients to become negative.

References

• arXiv:cea-01274195v1_document
• Li, X.-J. (1997). The positivity of a sequence of numbers and the Riemann hypothesis. Journal of Number Theory.
• Keiper, J. B. (1992). Power series expansions of Riemann's xi function. Mathematics of Computation.