Spectral Asymptotics and Conformal Geometry: A New Framework for the Riemann Hypothesis

Executive Summary

The technical paper arXiv:cea-03673957_file_txt_V1 introduces a transformative framework for addressing the Riemann Hypothesis (RH) by synthesizing analytic number theory with semiclassical spectral analysis. The central innovation involves a conformal transformation, x = 1/(1-z), which maps the critical strip of the Riemann zeta function into the unit disk in the complex z-plane. This mapping redefines the RH as a condition of analyticity for the logarithm of the completed zeta function, Φ(z), within the disk's interior.

A primary finding of the source is the asymptotic dichotomy of the Keiper-Li coefficients, λ_n. If the RH holds, these coefficients grow logarithmically; if it fails, the presence of off-critical zeros triggers exponential divergence. By establishing an analogy with the Schrödinger operator for a quartic oscillator, the paper suggests that the distribution of Riemann zeros can be modeled as a physical spectrum governed by Bohr-Sommerfeld quantization. This approach provides a rigorous methodology for quantifying the computational limits of RH verification, identifying that indices as high as n > 10⁶⁰ may be necessary to definitively detect violations at current verified heights.

Introduction

The Riemann Hypothesis remains the most profound challenge in analytic number theory, asserting that all non-trivial zeros of the zeta function ζ(s) possess a real part of 1/2. While traditional methods rely on prime counting functions and the explicit formula, recent research has shifted toward the Keiper-Li criterion. This criterion relates the truth of the RH to the positivity of a sequence of coefficients derived from the Taylor expansion of the completed zeta function.

In arXiv:cea-03673957_file_txt_V1, the author explores the deep mathematical structures underlying these coefficients. The motivation is to move beyond mere numerical observation toward a spectral understanding of the zeta function's zeros. By treating the zeros as the spectrum of a dynamical system, the paper leverages exact WKB (Wentzel-Kramers-Brillouin) methods to probe the behavior of the zeta function in the complex plane. This analysis contributes a new perspective on the "classical-action period" of the zeta function, suggesting that the zeros are not merely arithmetic accidents but are constrained by the same laws that govern quantum mechanical systems.

Mathematical Background

The foundation of this analysis is the completed zeta function, 2ξ(x), defined to satisfy the functional equation 2ξ(x) = 2ξ(1-x). The source paper utilizes the normalization:

2ξ(x) = x(x-1) π^-x/2 Γ(x/2) ζ(x)

The critical innovation is the composition of this function with the conformal map x = 1/(1-z). This map pulls the half-plane Re(x) > 1/2 into the unit disk |z| < 1. Consequently, the critical line Re(x) = 1/2 becomes the boundary of the unit circle |z| = 1. The function Φ(z) is then defined as:

Φ(z) = log 2ξ(1/(1-z))

The Keiper-Li coefficients, λ_n, are the Taylor coefficients of Φ(z) at z = 0. The RH is equivalent to the statement that Φ(z) is analytic for all |z| < 1. If a zero ρ exists off the critical line, its image z_ρ will fall inside the unit disk, creating a singularity that disrupts the analyticity of Φ(z) and causes the coefficients λ_n to fluctuate and eventually grow exponentially.

Main Technical Analysis

Spectral Properties and Zero Distribution

The paper arXiv:cea-03673957_file_txt_V1 demonstrates that dΦ/dz is a meromorphic function with simple poles at every point z_ρ corresponding to a Riemann zero. The residues at these poles are consistently 1. This allows the coefficients λ_n to be represented as a sum over zeros:

λ_n = ∑ [1 - (1 - 1/ρ)ⁿ]

When the RH is true, all zeros ρ have Re(ρ) = 1/2, meaning their images lie exactly on the circle |z| = 1. In this case, the sum is dominated by the density of the zeros on the critical line, resulting in the smooth asymptotic growth:

λ_n ∼ 0.5 n [log n + (γ - log 2π - 1)]

where γ is Euler's constant. This logarithmic trend is the "signature" of the Riemann Hypothesis being satisfied. Any deviation from the critical line would introduce a term (1 - 1/ρ)ⁿ where the base has a magnitude less than 1, leading to exponential growth that eventually overwhelms the logarithmic background.

Quantitative Obstructions and Index Bounds

A significant contribution of this work is the quantification of how difficult it is to detect a violation of the RH using these coefficients. The paper establishes that if a zero exists at height T with a real part σ > 1/2, its influence on λ_n is masked by the contributions of all other zeros until n reaches a critical threshold.

• Necessary Condition: To detect a zero at height T, n must be at least 0.5 T exp(1/t). For current verified heights of T ≈ 10¹³, this implies n must be at least 10¹³.
• Sufficient Condition: To guarantee that a violation is not missed, the bound is significantly stricter: n ≳ T^{1 + 2/t}. For current zero-free regions, this suggests n > 10⁶⁰.

This explains why previous numerical studies, which often only reached n = 10⁵ or 10⁶, were insufficient to provide definitive evidence for or against the hypothesis.

The Quartic Oscillator Analogy

The analysis draws a compelling parallel between the zeta function and the Schrödinger operator H = -d²/dq² + q⁴. The eigenvalues E_l of this operator follow a Bohr-Sommerfeld quantization rule that mirrors the distribution of Riemann zeros. The source paper suggests that λ_n can be viewed as spectral invariants of this system. The "classical-action period" associated with the potential q⁴ relates to the logarithmic distribution of primes, providing a physical mechanism for the spacing of the zeros.

Novel Research Pathways

1. Node-Sampled Residue Functionals

Formulation: Construct a test functional L[Φ] using a contour integral weighted by a rational function W(z) that has poles at specific nodes z_m on the unit circle.

Connection: By the residue theorem, L[Φ] becomes a linear combination of Φ(z_m). If the RH is false, an interior singularity will contribute a residue that is absent if the RH is true.

Methodology: Optimize the placement of nodes z_m to maximize the sensitivity to zeros in the region Re(s) > 1/2 while suppressing the "noise" from the known zeros on the critical line.

2. Hyperasymptotic Remainder Improvement

Formulation: Apply exact WKB methods to the remainder term R(n) = λ_n - AsymptoticValue.

Connection: The remainder R(n) contains information about the fine-scale distribution of primes. By expanding R(n) into a series of exponentially small corrections (Stokes phenomena), one can achieve higher precision in coefficient prediction.

Methodology: Use Borel summation to regularize the divergent series of 1/n terms in the λ_n expansion, potentially reducing the index n required for computational verification from 10⁶⁰ to a more manageable range.

Computational Implementation

The following Wolfram Language code calculates the Keiper-Li coefficients and compares them to the RH-true asymptotic baseline, while also mapping zeros to the z-plane disk.

(* Section: Spectral Coefficient and Z-Plane Analysis *)
(* Purpose: Compute lambda_n and visualize the conformal mapping of zeros *)

Module[{nMax, zeros, zMap, lambdaK, baseline, rho, data, n},
  nMax = 30;
  
  (* Define the conformal map z = 1 - 1/rho *)
  zMap[rho_] := 1 - 1/rho;
  
  (* Get the first 100 zeros on the critical line *)
  zeros = Table[1/2 + I * Im[ZetaZero[k]], {k, 1, 100}];
  
  (* Compute Keiper-Li coefficients using the sum over zeros formula *)
  lambdaK[n_] := Re[Total[Table[1 - (1 - 1/rho)^n, {rho, zeros}]] + 
                 Total[Table[1 - (1 - 1/Conjugate[rho])^n, {rho, zeros}]]];
  
  (* RH-true asymptotic baseline: 0.5 * n * (log n + gamma - log 2pi - 1) *)
  baseline[n_] := 0.5 * n * (Log[n] + EulerGamma - Log[2 * Pi] - 1);
  
  (* Generate comparison data *)
  data = Table[{n, lambdaK[n]/n, baseline[n]/n}, {n, 1, nMax}];
  
  (* Plot 1: Coefficient Growth Comparison *)
  Print[ListLinePlot[data[[All, {1, 2}]],
    PlotStyle -> Blue, 
    PlotLegends -> {"Computed lambda_n/n"},
    Frame -> True, 
    PlotLabel -> "Normalized Keiper-Li Coefficient Growth"]];
    
  (* Plot 2: Z-plane distribution of zeros *)
  Print[ListPlot[ReIm[zMap /@ zeros], 
    PlotStyle -> Red, 
    AspectRatio -> 1, 
    PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}}, 
    Epilog -> Circle[{0,0}, 1], 
    PlotLabel -> "Riemann Zeros Mapped to the Z-Plane Unit Disk"]];
]

Conclusions

The analysis of arXiv:cea-03673957_file_txt_V1 reveals that the Riemann Hypothesis is deeply encoded in the analyticity of the zeta function under conformal transformation. The Keiper-Li coefficients provide a rigorous, albeit computationally intensive, metric for testing this analyticity. The most promising research direction lies in the application of exact WKB methods to the spectral remainders of these coefficients, which could bypass the astronomical index requirements currently hindering numerical verification. By viewing the zeros through the lens of quantum oscillators, we move closer to a unified theory that links the distribution of prime numbers to the fundamental laws of spectral dynamics.

References

• arXiv:cea-03673957_file_txt_V1: Spectral Approach to the Riemann Hypothesis.
• 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.