Exploring the Spectral Geometry of Prime Numbers via Stationary Harmonic Sets

Introduction

The study of lacunary sets in harmonic analysis—specifically Sidon sets and their generalizations—has provided a rigorous framework for understanding the interplay between algebraic structure and probabilistic behavior. The research presented in arXiv:hal-00766023v1 extends this tradition by investigating stationary sets and the specific norms that govern their behavior in Banach spaces. While the original paper focuses on the properties of subsets of discrete groups and the unconditional convergence of Fourier series, its mathematical machinery offers profound implications for analytic number theory, particularly the distribution of the zeros of the Riemann zeta function.

The Riemann Hypothesis (RH) posits that all non-trivial zeros of the zeta function lie on the critical line where the real part is 1/2. Central to this investigation is the behavior of the zeta function on this line, often modeled as a random Dirichlet series. The lacunarity of the set of primes suggests that the Dirichlet kernels associated with the zeta function might share structural similarities with the stationary sets defined in the source paper. This article provides a technical bridge between the harmonic analysis of stationary sets and the spectral properties of the Riemann zeta function, exploring how the norm definitions and symmetrization techniques used to characterize Sidon-like behavior can be adapted to study the moments of the zeta function.

Mathematical Background

To establish the connection, we must first define the core objects from arXiv:hal-00766023v1. Let G be a compact abelian group and Gamma its dual group. A subset Lambda of Gamma is called a Sidon set if there exists a constant C such that for every finite Lambda-polynomial f, the sum of the absolute values of its Fourier coefficients is bounded by C times the uniform norm of f.

The source paper introduces a more general class of sets, known as stationary sets. These are characterized by the behavior of the unconditional norm, denoted as [[f]]. This norm is defined as the supremum of the absolute value of the integral over a probability space of the convolution of a random measure with a randomized version of the function f. Specifically, the paper establishes that for a subset Lambda, the condition for being a Sidon set is tied to whether the probabilistic point of view of almost sure convergence can be exchanged for a topological one.

A central result in arXiv:hal-00766023v1 is that for certain randomized polynomials R_N, the unconditional norm [[R_N]] is bounded by a constant K times 2 raised to the power of (N+1)/2. This square-root growth is a hallmark of random walks and suggests that stationary sets capture the essence of independence within structured frequency sets, a property highly relevant to the pseudo-random behavior of the zeta function on the critical line.

Symmetrization and Moment Estimates

The technical heart of arXiv:hal-00766023v1 involves a complex symmetrization process for functions. The function f is decomposed into a sum over distinct indices of products of characters and their conjugates. This expansion is strikingly similar to the expansion of the moments of the zeta function. When calculating the 2k-th moment of the zeta function on the critical line, one encounters sums over products of k primes and their inverses. The "distinct index" summation in the source paper corresponds to the "off-diagonal" terms in the moment expansions of L-functions.

If the set of logs of primes behaves as a stationary set, then the unconditional norm provides a bound on the fluctuations of the Dirichlet sums. The source paper's result regarding the bound of [[R_N]] suggests that for a stationary set, the random fluctuations are bounded by a square-root law. In the context of the Riemann Hypothesis, this relates to the fluctuations of the zero-counting function. If the spectrum of the zeta zeros satisfies the conditions of a stationary set, it would imply that the zeros are distributed with a regularity that prevents large clusters, a property known as repulsion.

Entropy Bounds and Zero Density

Another significant contribution of the source paper is the derivation of entropy-like bounds for the Fourier spectrum. It demonstrates that for all measures mu supported on a set Lambda, the number of characters where the Fourier transform of mu is greater than a threshold delta is bounded by the exponential of a constant times the norm of mu squared divided by delta squared. This inequality is a functional-analytic form of a large sieve inequality.

In analytic number theory, zero-density estimates bound the number of zeros with a real part greater than a fixed value. These estimates often rely on controlling the set of points where a Dirichlet polynomial is large. By mapping the entropy bounds from arXiv:hal-00766023v1 to the set of frequencies defined by the logarithms of primes, we can theoretically constrain the frequency of "large values" of the zeta function. Such constraints are essentially equivalent to proving that zeros cannot cluster too far away from the critical line, providing a pathway to refining zero-free regions.

Novel Research Pathways

Pathway 1: Scale-dependent Sidon Constants for Prime Logs

One promising direction is to determine if the set of logs of primes satisfies the definition of a stationary set within the group of real numbers. This involves investigating whether the constant K_s is finite for the unconditional norm defined in the source. Researchers could use the Hardy-Littlewood k-tuple conjecture to estimate the correlations of prime logs and check if they satisfy the symmetrization identities found in arXiv:hal-00766023v1. A proof of stationarity would strengthen existing central limit theorems for the zeta function.

Pathway 2: Unconditional Norms for Randomized Euler Products

Another pathway involves importing the unconditional norm to randomized Euler products. By defining a randomization where prime-frequency coefficients are multiplied by independent signs, one could attempt to prove inequalities of the form [[f]] is less than or equal to a constant times the uniform norm of f. If such a bound holds for Euler-product-supported functions, it would yield new sub-gaussian tail bounds for the supremum of the zeta function on the critical line, a key requirement for the Lindelof Hypothesis.

Pathway 3: Stationarity in Higher-Order Product Spaces

The source paper explores stationarity in k-fold products of groups. This can be translated to the study of higher moments of zeta. By treating the off-diagonal terms in the 2k-th moment expansion as components of a stationary set decomposition in a high-dimensional frequency space, researchers may identify new cancellation mechanisms that are currently invisible to standard sieve methods.

Computational Implementation

(* Section: Stochastic Fluctuations of Dirichlet Polynomials *)
(* Purpose: Illustrate Sidon-like growth in randomized Dirichlet sums *)

Module[{
  N = 300, 
  mZeros = 20, 
  zeros, 
  tGrid, 
  trials = 50, 
  valsRand, 
  avgSup
},
  (* Obtain ordinates of the first 20 non-trivial zeros *)
  zeros = Table[Im[ZetaZero[k]], {k, 1, mZeros}];
  
  (* Create a grid of t-values near these zeros *)
  tGrid = Flatten @ Table[zeros[[k]] + h, {k, 1, mZeros}, {h, {-0.1, 0, 0.1}}];
  
  (* Calculate the maximum of a randomized Dirichlet polynomial over trials *)
  (* This simulates the f^alpha randomization in arXiv:hal-00766023v1 *)
  valsRand = Table[
    Max[Table[
      Abs[Sum[RandomChoice[{-1, 1}] * n^(-1/2 - I * t), {n, 1, N}]], 
      {t, tGrid}
    ]], 
    {trials}
  ];
  
  avgSup = Mean[valsRand];
  
  Print["Mean randomized sup norm over sampled t-grid: ", NumberForm[avgSup, {6, 4}]];
  
  (* Visualize the distribution of the sup norms *)
  Histogram[valsRand, 15, "Probability", 
    PlotLabel -> "Distribution of Randomized Sup Norms", 
    AxesLabel -> {"Sup Norm", "Frequency"}]
]

Conclusions

The analysis of arXiv:hal-00766023v1 suggests that stationary sets occupy a unique position between the rigid structure of Sidon sets and the total randomness of generic subsets. By defining the unconditional norm and demonstrating its relationship to almost sure convergence, the source paper provides a robust framework for evaluating the "randomness" of arithmetic sequences. The connection to the Riemann Hypothesis lies in the interpretation of the primes—and the resulting zeros of the zeta function—as a stationary system. If the logs of primes are indeed stationary, the fluctuations of the zeta function on the critical line are not merely random but are constrained by specific constants that enforce stability in the zero distribution. Future work should focus on calculating these constants for the primes and determining if they satisfy the growth rates required for a zero-free region.

References

• arXiv:hal-00766023v1: Sidon sets and stationary sets in harmonic analysis.
• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function. Proc. Sympos. Pure Math.
• Bourgain, J. (1985). Sidon sets and probabilistic methods. Lecture Notes in Mathematics.