Harmonic Analysis of Stationary Sets and the Spectral Distribution of Zeta Zeros

Download Full Article

This article is available as a downloadable PDF with complete code listings and syntax highlighting.

Introduction

The Riemann Hypothesis (RH) remains the most profound challenge in analytic number theory, asserting that the non-trivial zeros of the Riemann zeta function, ζ(s), are restricted to the critical line Re(s) = 1/2. Modern strategies to resolve this conjecture often transcend traditional complex analysis, moving into the realm of harmonic analysis on compact groups and the spectral theory of operators. A central theme in this intersection is the study of Dirichlet polynomials as trigonometric polynomials on infinite-dimensional tori, a framework known as the Bohr lift.

The source paper arXiv:hal-00766023 provides a rigorous foundation for this approach by developing the theory of stationary sets and Sidon sets. These mathematical structures describe "thin" sets of characters in a dual group where Fourier coefficients exhibit near-independence. By applying the technical machinery of random measures and norm estimates presented in the paper, researchers can establish new bounds on the fluctuations of the zeta function. This article synthesizes the findings of the source paper to propose a harmonic-analytic bridge between the distribution of primes and the spectral properties of zeta zeros.

Mathematical Background

The core of the analysis in arXiv:hal-00766023 involves the dual group Γ of a compact abelian group G. A subset Λ of Γ is defined as a Sidon set if there exists a constant C such that for every trigonometric polynomial f supported on Λ, the sum of the absolute values of the Fourier coefficients is bounded by the supremum norm: sum |f(γ)| ≤ C ||f||_∞.

The paper introduces a more generalized object: the stationary set. This is defined through a specific seminorm, denoted in the source as [[ f ]], which is the supremum over a family of random measures μ_α. Specifically, the paper investigates the property where the norm of a randomized Fourier series is controlled by the L^∞ norm of the function itself. This is critical for the Riemann Hypothesis because the sequence of logarithms of primes {log p} behaves similarly to these thin sets when viewed through the Bohr lift.

Key theorems in the source paper establish that products of stationary sets remain stationary (Structure 13). This multiplicative stability is essential for analyzing the Euler product representation of ζ(s), which decomposes the function into a product over primes. If the set of primes satisfies the criteria for stationarity, the growth of ζ(s) on the critical line can be constrained by the square-root cancellation laws typical of random walks in Banach spaces.

Main Technical Analysis

Spectral Properties and Zero Distribution

A primary result of arXiv:hal-00766023 is the exponential counting bound for Fourier coefficients (Structure 15). For any measure μ supported on a thin set Λ, the number of characters where the Fourier transform exceeds a threshold δ is bounded by:

| { γ in Λ : |μ(γ)| ≥ δ } | ≤ exp( c ||μ||² / δ² )

This is a large-deviation inequality that has significant implications for the vertical distribution of zeta zeros. If the imaginary parts of the zeros are treated as a spectral set, this inequality limits the degree of clustering possible on the critical line. It suggests that the "spikiness" of the zeta function is naturally suppressed by the underlying group-theoretic structure of its frequencies.

Norm Estimates and the Lindelöf Hypothesis

The paper provides explicit bounds for the growth of random polynomials R_N (Structure 7), showing that [[ R_N ]] ≤ K_s(Λ) 2^(N+1)/2. In the context of the zeta function, this square-root growth mirrors the predicted behavior of the Möbius function sums, which is equivalent to the Riemann Hypothesis. The Lindelöf Hypothesis, which states that ζ(1/2 + it) grows slower than any power of t, can be reframed as a statement about the stationarity of the set of prime logarithms.

Furthermore, the source paper discusses the existence of majorizing measures (Structure 8). It proves that for any function h in L²(G), there exists a continuous function f that majorizes the Fourier coefficients of h while maintaining a controlled supremum norm. This majorization principle is a powerful tool for constructing extremal functions used to bound the number of zeros in short intervals on the critical line.

Random Measures and Unconditionality

The construction of measures μ as integrals over random perturbations (Structure 4) allows for a "derandomization" process. By representing a deterministic Dirichlet series as an average of randomized versions, researchers can inherit the stability of the random model. This provides a mechanism to prove that the zeta function cannot stay large for long intervals, as the "phases" p^-it eventually behave like independent random variables, leading to the cancellation effects described in the paper's analysis of Λ(p) sets (Structure 14).

Novel Research Pathways

Pathway 1: The Stationary Property of Prime Logarithms

Formulation: Investigate if the set Λ = {log p : p is prime} satisfies the stationary set criteria within the Bohr compactification of the real line. This involves testing if the norm [[ f ]] for prime-supported polynomials grows as the square root of the number of terms.

Methodology: Apply the random measure constructions from arXiv:hal-00766023 to truncated Euler products. If Λ is stationary, it would provide a functional-analytic proof of the Möbius randomness principle, which is a core component of the RH.

Pathway 2: Entropy Bounds as Constraints on Zero Localization

Formulation: Use the exponential counting bound (Structure 15) to develop an uncertainty principle for zeta zeros. This would relate the concentration of zeros in a spectral window to the total variation of a measure constructed from prime weights.

Methodology: Map the imaginary parts of zeta zeros to characters on a compact group. By applying the entropy inequalities from the source paper, we can determine the maximum possible density of zeros consistent with known bounds on prime counts.

Computational Implementation

Wolfram Language

(* Section: Random Prime Dirichlet Polynomial Analysis *)
(* Purpose: This code simulates a randomized prime sum to compare *)
(* its fluctuations against the stationary bounds in hal-00766023. *)

Module[{
  Tmin = 100, Tmax = 300, 
  P = 50, 
  primes, eps, a, tGrid, 
  dirichletPoly, zetaLog, 
  plt1, plt2, zeros, zeroOrds
},
  (* Get primes up to P *)
  primes = Prime[Range[PrimePi[P]]];
  
  (* Generate random Rademacher signs {-1, 1} *)
  SeedRandom[42];
  eps = RandomChoice[{-1, 1}, Length[primes]];
  a = AssociationThread[primes -> eps];
  
  (* Define randomized Dirichlet polynomial D(t) *)
  dirichletPoly[t_] := Sum[a[p] * p^(-1/2) * Exp[-I * t * Log[p]], {p, primes}];
  
  (* Calculate Zeta for comparison *)
  zetaLog[t_] := Log[Abs[Zeta[1/2 + I * t]]];
  
  (* Generate data points *)
  tGrid = Range[Tmin, Tmax, 0.2];
  dataD = Table[{t, Re[dirichletPoly[t]]}, {t, tGrid}];
  dataZ = Table[{t, zetaLog[t]}, {t, tGrid}];
  
  (* Identify Zeta Zeros in the range *)
  zeroOrds = Select[Table[Im[ZetaZero[n]], {n, 1, 100}], Tmin <= # <= Tmax &];
  
  (* Visualization *)
  plt1 = ListLinePlot[dataD, PlotStyle -> Blue, 
    PlotLabel -> "Randomized Prime Dirichlet Polynomial (Re Part)",
    AxesLabel -> {"t", "Re D(t)"}];
    
  plt2 = ListLinePlot[dataZ, PlotStyle -> Red, 
    PlotLabel -> "Log|Zeta(1/2 + it)|",
    AxesLabel -> {"t", "Log|Zeta|"}];
    
  (* Mark zeros as points on the x-axis of the Dirichlet plot *)
  Show[plt1, Graphics[{Red, PointSize[Medium], 
    Point[Table[{z, 0}, {z, zeroOrds}]]}], 
    PlotRange -> All]
]

Conclusions

The mathematical structures established in arXiv:hal-00766023 offer a sophisticated toolkit for probing the Riemann Hypothesis through the lens of harmonic analysis. By treating the distribution of primes and zeta zeros as problems of stationarity and Sidon-type independence, we can derive rigorous bounds that align with the expected behavior of the zeta function on the critical line. The most promising avenue for further research lies in the formal characterization of the set of prime logarithms as a stationary set, which would provide a structural explanation for the square-root cancellation required by the RH. Future steps should focus on quantifying the constants K_s(Λ) for prime-related sets and applying the exponential counting bounds to refine zero-density estimates.

References

arXiv:hal-00766023: Stationary sets, Sidon sets and their properties in harmonic analysis.
Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function.
Bohr, H. (1924). Zur Theorie der fastperiodischen Funktionen.
Soundararajan, K. (2009). Moments of the Riemann zeta function.