The Li Criterion and the Positivity Buffer: A Sequence-Based Approach to the Riemann Hypothesis

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Introduction

The Riemann Hypothesis remains one of the most profound unsolved problems in mathematics, asserting that all non-trivial zeros of the Riemann zeta function, zeta(s), lie on the critical line where the real part is exactly 1/2. While traditional approaches focus on the complex analysis of the zeros themselves, the paper arXiv:hal-04588365v1 investigates a compelling alternative: the Li criterion. This criterion transforms the problem into a question of the positivity of a real-valued sequence known as Keiper-Li coefficients.

By shifting the perspective from the complex plane to a sequence of real numbers, this approach offers new tools for both theoretical analysis and computational verification. The core idea is that the distribution of zeros is encoded in the growth of these coefficients. This article examines how the arithmetic properties of the zeta function create a buffer that ensures the positivity of these coefficients, provided no zeros deviate from the critical line.

Mathematical Background

The primary object in this framework is the completed zeta function, xi(s), which is an entire function that satisfies the symmetric functional equation xi(s) = xi(1-s). The Li criterion involves a conformal mapping that sends the half-plane Re(s) > 1/2 to the interior of the unit disk. Under this mapping, the Riemann Hypothesis is equivalent to the statement that all zeros of the transformed function lie on the boundary of the unit circle.

The Keiper-Li coefficients, lambda_n, are defined by the power series expansion of the logarithmic derivative of this transformed function. They can be explicitly related to the non-trivial zeros rho by the sum: lambda_n = sum(1 - (1 - 1/rho)ⁿ). The Riemann Hypothesis holds if and only if lambda_n > 0 for every positive integer n.

Main Technical Analysis

Arithmetic Growth and the Positivity Buffer

The analysis in arXiv:hal-04588365v1 demonstrates that the coefficients lambda_n can be decomposed into an arithmetic part and a zero-dependent part. The arithmetic part is determined by fundamental constants, including the Euler-Mascheroni constant and the values of the zeta function at integers. It is shown that this arithmetic component grows at a rate of approximately (n/2) log n.

This growth is essential because it acts as a "positivity buffer." If a zero were to exist off the critical line, its contribution to the sum would grow exponentially, eventually overcoming the arithmetic growth and causing lambda_n to become negative for sufficiently large n. The paper argues that the known density of zeros on the critical line prevents such a sign change from occurring.

The Li Inequality and Zero Density

A significant portion of the technical analysis involves bounding the fluctuations of the zero-dependent sum. By applying the Guinand-Weil explicit formula, the researcher establishes that the oscillatory contributions from the zeros are insufficient to overcome the leading logarithmic term. The derived asymptotic behavior, lambda_n = (n/2) log n + C*n + O(sqrt(n) log n), provides a quantitative basis for the continued positivity of the sequence.

Novel Research Pathways

Complete Monotonicity: A promising avenue is to investigate whether lambda_n is a completely monotonic sequence. This would imply a deeper structural rigidity in the zero distribution than mere positivity, potentially linking the problem to the theory of positive definite functions.
Selberg Class Generalization: The methods used for the Riemann zeta function can be extended to the broader Selberg class of L-functions. This would involve calculating the "positivity buffer" for higher-degree L-functions to verify the Generalized Riemann Hypothesis.
P-adic Interpolation: Exploring p-adic analogs of the Li coefficients could connect the analytic behavior of the zeta function to the algebraic structures of cyclotomic fields and Iwasawa theory.

Computational Implementation

The following Wolfram Language code provides a framework for evaluating the Keiper-Li coefficients numerically by summing over the first 1000 non-trivial zeros and comparing the result to the arithmetic trend predicted in arXiv:hal-04588365v1.

(* Section: Numerical Evaluation of Li Coefficients *)
(* Purpose: Compute lambda_n using Riemann zeros and verify positivity *)

Module[{numZeros = 1000, maxN = 50, zeros, lambdaValues, approx, n},
  
  (* 1. Obtain the ordinates of the first 1000 non-trivial zeros *)
  zeros = Table[1/2 + I*ZetaZero[k], {k, 1, numZeros}];
  
  (* 2. Define the Li coefficient function lambda_n *)
  (* lambda_n = sum over zeros [1 - (1 - 1/rho)^n] *)
  lambda[n_] := Total[Table[
    Re[ (1 - (1 - 1/r)^n) + (1 - (1 - 1/(1 - r))^n) ], 
    {r, zeros}
  ]];

  (* 3. Generate a table of coefficients *)
  lambdaValues = Table[{n, lambda[n]}, {n, 1, maxN}];
  
  (* 4. Approximation from the arithmetic part *)
  (* S_n approx (n/2) Log[n] + n/2 (EulerGamma - 1 - Log[2 Pi]) *)
  approx[n_] := (n/2)*Log[n] + (n/2)*(EulerGamma - 1 - Log[2*Pi]);
  
  (* 5. Output results *)
  Print["First 10 Li Coefficients:"];
  Print[TableForm[Take[lambdaValues, 10], TableHeadings -> {None, {"n", "lambda_n"}}]];
  
  ListPlot[{
    lambdaValues, 
    Table[{n, approx[n]}, {n, 1, maxN}]},
    PlotLegends -> {"Calculated (1000 zeros)", "Arithmetic Trend"},
    PlotLabel -> "Growth of Li Coefficients lambda_n",
    AxesLabel -> {"n", "lambda_n"},
    Joined -> True,
    PlotStyle -> {Blue, {Red, Dashed}}
  ]
]

Conclusions

The study of arXiv:hal-04588365v1 highlights the Li criterion as a powerful reformulative tool for the Riemann Hypothesis. By shifting the focus from complex zeros to a real-valued sequence, the research reveals the delicate balance between arithmetic constants and zero distribution. The primary conclusion is that the logarithmic growth of the coefficients provides a robust defense of the hypothesis, and the next step for research should focus on refining the error terms in the coefficient expansion to ensure this buffer holds for all n.

References

arXiv:hal-04588365v1 - A new approach to the Riemann hypothesis.
Li, X.-J. (1997). The Positivity of a Sequence of Numbers and the Riemann Hypothesis. Journal of Number Theory, 65(2), 325-333.
Bombieri, E., and Lagarias, J. C. (1999). Complements to Li's Criterion for the Riemann Hypothesis. Journal of Number Theory, 77(2), 274-287.