Conformal Mapping and Spectral Asymptotics: A New Lens on the Riemann Hypothesis

Executive Summary

The research paper arXiv:cea-03673957v1_document_document presents a sophisticated reformulation of the Riemann Hypothesis (RH) by utilizing a conformal mapping that translates the distribution of zeta zeros into a problem of analyticity within the unit disk. By mapping the half-plane Re(x) > 1/2 to the disk |z| < 1 via the transformation x = 1/(1-z), the study demonstrates that RH is equivalent to the analyticity of the logarithmic completed zeta function. The core insight involves the Keiper-Li coefficients, which exhibit logarithmic growth under RH but exponential growth if zeros exist off the critical line. This approach is particularly promising because it introduces a "split-pole" residue method to stabilize numerical computations and draws deep analogies between number theory and the spectral properties of Schrödinger operators with polynomial potentials.

Introduction

The Riemann Hypothesis remains one of the most profound challenges in mathematics, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. While numerical evidence supports the hypothesis for trillions of zeros, a rigorous proof requires a deeper understanding of the function's growth and the distribution of its zeros in the complex plane. The paper arXiv:cea-03673957v1_document_document approaches this problem by leveraging the Keiper-Li criterion, which translates the location of zeros into the behavior of a discrete sequence of real numbers.

Traditional methods for calculating these coefficients often suffer from numerical instability due to high-order differentiation. The source paper addresses this by introducing a generalized construction of the sequence using residue calculus and conformal deformations. By mapping the right-hand side of the critical strip into the unit disk, any zero ρ with Re(ρ) > 1/2 corresponds to a singularity inside the disk, causing the Taylor coefficients to diverge exponentially. This article explores the mathematical structures of this transformation and proposes research directions that bridge analytic number theory with spectral analysis.

Mathematical Background

The foundation of this analysis rests upon the completed zeta function, 2ξ(x), defined as the product of x(x-1), π raised to -x/2, the gamma function Γ(x/2), and the Riemann zeta function ζ(x). This function is entire and symmetric around the critical line Re(x) = 1/2. The study introduces the function Φ(z) defined as the logarithm of 2ξ(x) where x is replaced by the conformal map 1/(1-z).

The mapping x = 1/(1-z) is a Möbius transformation with several critical properties. First, the point x = 1 maps to z = 0. Second, the critical line Re(x) = 1/2 maps to the unit circle |z| = 1. Third, the half-plane Re(x) > 1/2 maps to the interior of the unit disk |z| < 1. Consequently, the Riemann Hypothesis is equivalent to the statement that Φ(z) is analytic throughout the open unit disk.

The Keiper-Li coefficients λ_n are the Taylor coefficients of Φ(z) expanded around the origin. By Cauchy's integral formula, these coefficients are extracted via a contour integral around a small circle. If a zero exists off the critical line, Φ(z) possesses a singularity inside the disk, which forces the sequence λ_n to eventually become negative and grow exponentially in magnitude, providing a clear signal of an RH violation.

Main Technical Analysis

Spectral Properties and Zero Distribution

The distribution of Riemann zeros is analyzed through a spectral lens, drawing parallels to the discrete spectrum of a Schrödinger operator. The paper establishes that the sequence derived from the Keiper-Li coefficients satisfies a fundamental positivity criterion: RH is true if and only if there exists an index n₀ such that the sequence remains positive for all n > n₀. This criterion emerges from the spectral analysis of the operator -d²/dq² + q⁴ on the real line.

The eigenvalues of this operator follow an asymptotic distribution related to the classical action period. The paper suggests that the Riemann zeros are not merely number-theoretic values but are governed by the same semiclassical physics that dictates the energy levels of anharmonic oscillators. This spectral connection manifests through the relationship between eigenvalues and the zeros via the conformal transformation, where zeros on the critical line preserve the positivity of the sequence, while off-line zeros introduce negative contributions that dominate at large n.

Residue Calculus and the Split-Pole Method

A significant technical contribution of the paper is the "split-pole" method. Traditional Keiper-Li calculations involve a pole of order n+1 at z = 0, which is numerically ill-conditioned. The source proposes replacing this high-order pole with a product of simple poles at distinct points z₁, z₂, ..., z_n. This transforms the coefficient extraction into a weighted sum of function values Φ(z_m).

This discretization allows for a more stable probing of the function Φ(z) deeper into the unit disk. By choosing nodes adaptively, researchers can improve numerical conditioning and potentially amplify the signal of off-critical zeros. This method effectively increases the resolution of the search for singularities without the precision loss associated with high-order derivatives.

Growth Rates and RH Sensitivity

The paper provides a rigorous comparison of growth rates. Under the assumption that RH is true, the coefficients λ_n grow slowly and logarithmically, following the trend (1/2)[log n + (γ - log 2π - 1)]. However, if RH is false, the asymptotic behavior is dominated by the zero closest to the origin in the z-plane, leading to exponential growth. The paper notes that for a zero at height T, the coefficient index n must be extremely large (approximately 10¹³ or higher) to capture this divergence, explaining why numerical counterexamples are so elusive.

Novel Research Pathways

Optimization of Node Placement

One promising direction is the optimization of the "split-pole" node set. By choosing nodes that approximate an optimal test kernel in the Hardy space of the disk, sensitivity to interior singularities can be maximized. Researchers could investigate node sets that minimize the amplification of roundoff error while probing specific arcs of the unit circle where high-frequency zeros are expected.

L-function Generalizations

The spectral reformulation naturally extends to other L-functions, suggesting a universal framework for understanding zero distributions. Applying this conformal mapping to Dirichlet L-functions or automorphic L-functions could reveal universal features of zero distributions across different arithmetic contexts. A specific program would focus on establishing uniform bounds for spectral coefficients across L-function families.

Quantum Chaos and Arithmetic Unique Ergodicity

The connection between Riemann zeros and quantum spectra opens pathways to quantum chaos theory. Investigating two-point correlation functions of the Keiper-Li coefficients and their connection to moments of the zeta function could provide new insights. This pathway could lead to a "quantum ergodicity" framework for arithmetic functions, potentially leading to probabilistic approaches to proving RH.

Computational Implementation

(* Section: Keiper-Li Coefficient Analysis *)
(* Purpose: Calculate lambda_n and verify logarithmic growth vs exponential risk *)

ClearAll["Global`*"];

(* Define the completed zeta function 2*xi(s) *)
xi[s_] := s (s - 1) * Pi^(-s/2) * Gamma[s/2] * Zeta[s];

(* Define the conformal map x = 1/(1-z) and the function Phi(z) *)
(* We use a series expansion of log(2*xi(1/(1-z))) around z=0 *)
phi[z_] := Log[2 * xi[1/(1 - z)]];

(* Extraction of Taylor coefficients lambda_n for n=1 to maxN *)
maxN = 40;
coeffs = CoefficientList[Series[phi[z], {z, 0, maxN}], z];

(* Extract lambda_1 to lambda_maxN (lambda_0 is log(1)=0) *)
lambdaValues = Table[N[coeffs[[n + 1]], 40], {n, 1, maxN}];

(* Theoretical growth if RH is true *)
rhTrueAsymptotic[n_] := 0.5 * (Log[n] + EulerGamma - Log[2*Pi] - 1);

(* Create a comparison table and plot *)
results = Table[{n, Re[lambdaValues[[n]]], rhTrueAsymptotic[n]}, {n, 1, maxN}];

Print[Grid[Prepend[results, {"n", "lambda_n", "Asymptotic"}], Frame -> All]];

ListLinePlot[{
  Table[{n, Re[lambdaValues[[n]]]}, {n, 1, maxN}], 
  Table[{n, rhTrueAsymptotic[n]}, {n, 1, maxN}]},
  PlotLegends -> {"Calculated lambda_n", "RH-True Trend"},
  PlotStyle -> {Blue, Red},
  AxesLabel -> {"n", "Value"},
  PlotLabel -> "Keiper-Li Coefficient Growth Analysis"
]

Conclusions

The analysis of arXiv:cea-03673957v1_document_document demonstrates that the Riemann Hypothesis is inextricably linked to the analyticity of the transformed function Φ(z) within the unit disk. By shifting the focus from the zeros themselves to the Taylor coefficients λ_n, the paper provides a discrete, real-valued metric for the RH. The transition from a single-pole residue to a distributed pole sum offers a significant improvement in numerical stability, which is essential for probing the high-frequency behavior of the zeta function.

The most promising avenue for further research is the integration of the spectral properties of the quartic potential with the Keiper-Li sequence. If the coefficients can be shown to correspond to the trace of a positive-definite operator, the positivity of λ_n would be established. Future work should focus on refining the bounds for the index n, particularly by exploring different conformal origins to reduce the computational complexity of verifying the RH at extreme heights.

References

• arXiv:cea-03673957v1_document_document
• Keiper, J. B. (1992). Power series expansions of log(ζ(s)). Mathematics of Computation, 58(198), 765-773.
• Li, X.-J. (1997). The positivity of a sequence of numbers and the Riemann hypothesis. Journal of Number Theory, 65(2), 325-333.