The Positivity Principle: Li's Criterion and the Spectral Stability of Zeta Zeros

Introduction

The Riemann Hypothesis remains the most profound unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. While the hypothesis is typically framed in the context of complex analysis and prime number distribution, the work presented in arXiv:hal-00654416v1 explores an equivalent formulation known as Li's criterion.

This criterion, established by Xian-Jin Li in 1997, translates the analytic behavior of the zeta function into the positivity of a discrete sequence of real numbers λ_n. The significance of the analysis in arXiv:hal-00654416v1 lies in its bridging of analytic number theory and functional analysis. By examining the arithmetic properties of these coefficients, researchers can investigate the distribution of the zeta function's zeros through a lens that resembles the Weil positivity in the theory of L-functions.

Mathematical Background

To understand the results in arXiv:hal-00654416v1, one must first define the completed Riemann zeta function, denoted as ξ(s). This function is defined to satisfy a clean functional equation and to remove the poles and trivial zeros of the standard ζ(s). The functional equation for this function is ξ(s) = ξ(1 - s), which implies a symmetry around the critical line Re(s) = 1/2. The non-trivial zeros ρ of ζ(s) are precisely the zeros of ξ(s).

Li's criterion involves the transformation of the complex variable s into a new variable z via the conformal mapping z = (s - 1) / s. This mapping takes the half-plane Re(s) > 1/2 to the unit disk |z| < 1. Crucially, it maps the critical line Re(s) = 1/2 exactly onto the unit circle |z| = 1. The Li coefficients λ_n are defined as the coefficients of the Taylor expansion of the logarithm of the ξ function after this change of variables. Li's Theorem states that the Riemann Hypothesis is true if and only if λ_n > 0 for all positive integers n.

Spectral Properties and Zero Distribution

The core of the technical argument in arXiv:hal-00654416v1 revolves around the representation of λ_n as a sum over the non-trivial zeros ρ of the zeta function. Using the Hadamard product, one can derive the following explicit formula for the Li coefficients: λ_n = ∑ [ 1 - ( (ρ - 1) / ρ )ⁿ ].

If the Riemann Hypothesis is true, then all zeros lie on the critical line. For a pair of zeros on the critical line, the terms in the sum can be grouped to show they are always non-negative. Conversely, if a zero exists off the critical line, there is a corresponding term with magnitude greater than 1. As n increases, this term will dominate the sum, causing λ_n to grow exponentially in the negative direction, eventually making λ_n < 0.

The paper arXiv:hal-00654416v1 emphasizes that λ_n can also be expressed using the Stieltjes constants and the values of the zeta function. Under the assumption of the Riemann Hypothesis, the coefficients grow according to the asymptotic rate: λ_n ~ (n / 2) log n - (n / 2) (1 + log 2π). This logarithmic growth is a signature of the density of the zeros on the critical line.

Novel Research Pathways

Localized Li Criterion for L-functions

A novel investigation would involve defining Local Li Coefficients associated with the Euler factor at a prime p. One would decompose the global λ_n into a sum of local contributions plus a contribution from the archimedean factor. This would involve studying the p-adic analog of the conformal mapping z = (s-1)/s to see if local positivity implies global stability.

Deformation of the Li Sequence via Volterra Operators

We propose a deformation of the sequence: λ_n(ε) = ∑ [ 1 - ( (ρ - 1) / ρ )ⁿ ] exp(-ε |ρ|). By introducing a regularization parameter ε, one can study the analytic continuation of the Li sequence. This allows for the use of Volterra-type integral operators to model the transition of λ_n as ε approaches zero, potentially revealing a differential equation governing zero stability.

Arakelov-Theoretic Interpretation

Building on arithmetic geometry, a promising direction is to interpret λ_n as a sequence of heights of certain cycles in an arithmetic surface. Using the framework of Arakelov theory, one can associate the completed zeta function with the determinant of a Laplacian on an arithmetic line bundle. If λ_n represents the volume of a convex body in an appropriate Euclidean space, its positivity would follow from fundamental geometric inequalities.

Computational Implementation

To visualize the behavior of the Li coefficients and verify the claims regarding their growth and positivity, we provide a Wolfram Language implementation. This code calculates λ_n using a finite number of known Riemann zeros.

(* Section: Li Coefficient Calculation *)
(* Purpose: Computes lambda_n using finite zeros to test positivity *)
Module[{numZeros = 150, nMax = 40, zeros, wValues, lambdaN, plot},
  zeros = Table[1/2 + I*ZetaZero[k], {k, 1, numZeros}];
  wValues = Map[(# - 1)/# &, zeros];
  lambdaN = Table[Re[Sum[1 - w^n, {w, wValues}]] * 2, {n, 1, nMax}];
  plot = ListLinePlot[lambdaN, 
    PlotStyle -> Blue, 
    PlotLabel -> "Growth of Li Coefficients", 
    AxesLabel -> {"n", "lambda_n"}];
  Print[plot];
]

Conclusions

The exploration of the Li criterion as detailed in arXiv:hal-00654416v1 provides a compelling alternative perspective on the Riemann Hypothesis. By transforming a question about the location of zeros in the complex plane into a question about the positivity of a real-valued sequence, Li's criterion opens the door to techniques from discrete mathematics and spectral theory.

The most promising avenue for further research appears to be the geometric interpretation via Arakelov theory. If the λ_n sequence can be rigorously linked to the curvature or capacity of arithmetic objects, the Riemann Hypothesis might finally be understood as a requirement for the geometric stability of the integers. Future steps should focus on the refinement of local Li coefficients and their behavior across general L-functions.

References

• arXiv:hal-00654416v1 - Source paper on the Li criterion
• Li, X.-J. (1997). The Positivity of a Sequence of Numbers and the Riemann Hypothesis. Journal of Number Theory, Vol. 65.
• Bombieri, E., and Lagarias, J. C. (1999). Complements to Li's Criterion for the Riemann Hypothesis. Journal of Number Theory, Vol. 77.