Spectral Geometry of Orthogonal Approximations and the Critical Line

Executive Summary

The research presented in arXiv:2006.02953 develops a sophisticated framework for approximating characteristic functions using orthogonal systems, establishing a rigorous link between the rate of convergence of approximation errors and the distribution of non-trivial zeros of the Riemann zeta function. By refining the Nyman-Beurling-Baez-Duarte (NBBD) criterion, the paper demonstrates that the L²-approximation error, denoted as D_n², can be transformed into a weighted integral on the critical line (σ = 1/2). This conversion is achieved through a Mellin-Plancherel identity that involves ζ(s), Gamma factors, and a specific basis of polynomials. The spectral properties of the resulting Gram matrices, particularly the shifting properties of matrices G₁ and G₂, encode information about zero-free regions of the zeta function. This approach is highly promising as it translates a problem of analytic number theory into one of operator theory and asymptotic linear algebra, offering a computational pathway to investigate the Riemann Hypothesis through the numerical analysis of matrix eigenvalue distributions.

Introduction

The Riemann Hypothesis (RH) asserts that the non-trivial zeros of the Riemann zeta function ζ(s) all lie on the critical line where the real part of s is 1/2. One of the most conceptually elegant reformulations of this problem is the Nyman-Beurling criterion, which characterizes the hypothesis in terms of the density of specific function spaces in L²(0, ∞). Traditionally, this involves approximating the indicator function χ(x) of the interval (0, 1] using linear combinations of fractional part functions. However, the utility of this criterion has long been limited by the difficulty of producing effective approximants and understanding the quantitative relationship between approximation error and zero distribution.

In arXiv:2006.02953, this program is advanced through the introduction of flexible multiplicative transforms and structured families of functions, g_k and h_k. By studying the approximation of target functions φ by these combinations, the authors establish that the failure of approximation is precisely controlled by the analytic behavior of a Mellin multiplier that incorporates the zeta function. This analysis provides a bridge between the Hilbert space geometry of functional approximation and the spectral properties of ζ(s). The contribution of this work lies in establishing explicit connections between matrix spectral theory and zero-free strips, providing both theoretical depth and a framework for numerical verification.

Mathematical Background

The fundamental mathematical object in this analysis is the Mellin transform, which serves as the multiplicative analogue of the Fourier transform. For a function f on (0, ∞), the transform is defined as the integral of f(x) multiplied by x raised to the power (s-1). The Mellin-Plancherel theorem allows us to identify L² norms on the positive real line with L² norms on a vertical line in the complex plane, specifically the critical line Re(s) = 1/2.

A key operator utilized in the source paper is the multiplicative transform g^×(t), which is defined as the integral of the fractional part of (x/t) multiplied by g(x)/x. This operator is significant because its Mellin transform intertwines with the zeta function. Specifically, the transform of the fractional part function {1/x} is related to -ζ(s)/s. Consequently, the approximation error D_n² in the physical space corresponds to a weighted distance in the Mellin space. The weight is determined by the magnitude of ζ(s) on the critical line, meaning that any zeros of the zeta function act as obstructions to the approximation.

Furthermore, the paper utilizes explicit Beta-function identities to compute transforms of basis functions. For instance, the integral of x^(s-1) divided by (1+x)^k from 0 to infinity is given by (Γ(s) Γ(k-s)) / Γ(k). These identities allow for the construction of basis functions whose Mellin transforms result in a polynomial P_k-1(s) divided by a Gamma factor, turning the problem into one of weighted polynomial approximation.

Main Technical Analysis

Spectral Properties and the Gram Matrix Mechanism

The core of the technical analysis in arXiv:2006.02953 centers on the Gram matrix G, where entries G_kj are the inner products of the basis functions. The paper demonstrates that these inner products satisfy a structured recurrence relation: <g_k^×, g_j^×> + <g_{k+1}^×, g_{j-1}^×> = (1 - r_k - r_{j-1}) <g_k^×, g_{j-1}^×>. This recurrence reveals that the Gram matrix possesses a near-Hankel structure, which is vital for understanding its spectral distribution.

The authors introduce modified Gram matrices G₁ and G₂ to represent shifting properties within the function space. The eigenvalues of these matrices are intimately related to the distribution of zeta zeros. If the Riemann Hypothesis holds, the eigenvalues of these matrices should satisfy specific asymptotic growth rates. Conversely, the existence of a zero off the critical line (Re(s) > 1/2) would manifest as a distinct growth signature in the spectral differences of these matrices. This provides a computational criterion: by monitoring the growth rate of spectral differences in increasingly large Gram matrices, one can detect the presence of zeros that violate the hypothesis.

Approximation Error and Quantitative Bounds

The source paper provides rigorous bounds for the approximation error by decomposing it into local and tail components. A critical parameter in this analysis is M_n, a truncation or scaling factor that grows with the size of the basis n. The paper establishes inequalities showing that the global approximation in L² is constrained by the local behavior of the functions at infinity. Specifically, if the functions G_n(x) satisfy certain decay conditions, the vanishing of the L² norm directly implies the non-vanishing of ζ(s) in the strip 1/2 < σ ≤ σ₀.

One of the most significant results is the identity expressing D_n² as the integral from negative infinity to infinity of |1/(s φ(s)) + ∑ (c_k,n/Γ(k)) P_k-1(s)|² multiplied by the weight |φ(s)|². In this context, φ(s) includes the zeta function as a multiplicative factor. Because the weight vanishes at the zeros of ζ(s), the function being approximated (1/φ(s)) becomes singular at those points. Thus, the feasibility of achieving a small approximation error is entirely dependent on whether these singularities exist to the right of the critical line.

Novel Research Pathways

1. Spectral Gap and Stability Analysis: A promising direction involves studying the smallest eigenvalue and the condition number of the Gram matrices. If the Riemann Hypothesis is true, the Gram matrices should remain relatively stable. If it is false, the intrusion of singularities into the approximation domain should cause the condition number to grow super-polynomially. Proving deterministic inequalities for these eigenvalues could lead to a new class of RH-sensitive stability theorems.

2. Generalization to Selberg Class L-functions: The framework in arXiv:2006.02953 is not limited to the Riemann zeta function. By replacing ζ(s) with a general L-function and adjusting the kernel functions, one could establish similar density criteria for the Generalized Riemann Hypothesis (GRH). This would suggest that functional density in L² is a universal property of automorphic forms.

3. Stochastic Approximation Models: The source paper mentions a probabilistic model where basis functions are defined as expectations of random variables. Investigating the variance of approximation errors under different probability distributions (such as Gamma or Log-normal) could provide a probabilistic proof of convergence, linking the deterministic properties of ζ(s) to the concentration of measure in high-dimensional Hilbert spaces.

Computational Implementation

The following Wolfram Language code demonstrates the construction of a Nyman-Beurling type Gram matrix and the subsequent projection of the target function to calculate the approximation error. This provides a numerical window into the spectral properties discussed in the source paper.

(* Section: Gram Matrix Projection and Error Analysis *)
(* Purpose: Demonstrate numerical projection of the indicator function *)
(* onto a fractional-part basis and compute the L2 error. *)

Module[
  {chi, basis, thetas, n, xmax, gram, bvec, coeffs, approx, err2, zeros},

  (* Define the target indicator function chi on (0,1] *)
  chi[x_] := Boole[0 < x <= 1];

  (* Set basis size and generate theta parameters *)
  n = 8;
  thetas = Table[k/(n + 1.0), {k, 1, n}];
  
  (* Define the basis of fractional part functions *)
  basis[k_][x_] := FractionalPart[thetas[[k]]/x];

  (* Define numerical integration range (truncation) *)
  xmax = 50;

  (* Construct the Gram Matrix G where G_jk = <basis_j, basis_k> *)
  gram = Table[
    NIntegrate[basis[j][x] * basis[k][x], {x, 0, xmax},
      Method -> "GlobalAdaptive", 
      WorkingPrecision -> 25, 
      AccuracyGoal -> 12
    ],
    {j, 1, n}, {k, 1, n}
  ];

  (* Construct the vector b where b_k = <chi, basis_k> *)
  bvec = Table[
    NIntegrate[chi[x] * basis[j][x], {x, 0, xmax},
      Method -> "GlobalAdaptive", 
      WorkingPrecision -> 25
    ],
    {j, 1, n}
  ];

  (* Solve the linear system G * c = b for coefficients c *)
  coeffs = LinearSolve[gram, bvec];

  (* Define the resulting approximation function *)
  approx[x_] := Sum[coeffs[[k]] * basis[k][x], {k, 1, n}];

  (* Calculate the L2 error squared *)
  err2 = NIntegrate[(chi[x] - approx[x])^2, {x, 0, xmax}];

  (* Output results and heights of first few Zeta zeros for context *)
  Print["Optimal Coefficients: ", coeffs // N];
  Print["L2 Approximation Error: ", err2 // N];
  Print["First 5 Zeta Zero Heights (Im): ", Table[Im[ZetaZero[k]], {k, 1, 5}] // N];

  (* Plot the target vs the approximation *)
  Plot[{chi[x], approx[x]}, {x, 0.05, 5}, 
    PlotRange -> {-0.2, 1.2}, 
    PlotLegends -> {"Target", "Approximation"},
    PlotLabel -> "Nyman-Beurling Approximation via Gram Matrix"]
]

Conclusions

The analysis of arXiv:2006.02953 demonstrates that the Riemann Hypothesis is deeply embedded in the structural properties of Hilbert spaces and the spectral geometry of their associated Gram matrices. By moving beyond simple dilatations and employing generalized kernels, the authors provide a robust framework for evaluating the density of these spaces in L²(0, ∞). The derived recurrence relations and explicit error bounds offer a clear quantitative target for a potential proof of the hypothesis.

The most promising avenue for further research is the spectral analysis of shifted matrices. If it can be shown that the relationship between these matrices precludes certain functional dependencies, the non-vanishing of ζ(s) in the critical strip would follow as a geometric necessity. Future steps should focus on refining the numerical stability of these calculations for much larger basis sets and investigating the universal properties of these Gram matrices across different classes of L-functions.

References

• arXiv:2006.02953: Riemann Hypothesis Connections via Orthogonal Approximations.
• arXiv:1805.06733: Generalized Nyman-Beurling Criterion and Admissible Target Functions.
• Baez-Duarte, L. (2003). A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis.
• Burnol, J.-F. (2002). L'extension de la transformation de Fourier-Beurling et la structure des espaces de Sonine.