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Finite Fourier Constraints and the Spectral Rigidity of Zeta Zeros

This article explores the connection between the discrete uncertainty principle on finite abelian groups, as detailed in arXiv:hal-00466459, and the distribution of zeros of the Riemann zeta function, proposing new pathways for bounding moments and zero-free regions through finite harmonic analysis.

January 06, 2026

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Introduction

The study of harmonic analysis on finite abelian groups has emerged as a critical laboratory for testing conjectures in analytic number theory. Central to this intersection is the Uncertainty Principle, a fundamental constraint that limits the simultaneous localization of a function and its Fourier transform. In the source paper arXiv:hal-00466459, the authors develop refined lower bounds for the support of the Fourier transform in groups such as Z/pZ, Z/p^2Z, and product groups Z/pZ x Z/qZ. While the paper focuses on combinatorial and algebraic properties, its implications reach into the heart of the Riemann Hypothesis (RH).

The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function, zeta(s), lie on the critical line Re(s) = 1/2. This conjecture is inherently spectral; it suggests a rigid structure in the distribution of primes, which can be viewed as the support of a distribution whose spectrum is defined by the zeros. The analysis in arXiv:hal-00466459 concerning the function theta(k, G)—defined as the minimum size of the support of the Fourier transform for a function with support size k—provides a discrete analogue to the density of zeros in the critical strip.

This article connects the discrete bounds of arXiv:hal-00466459 to the continuous spectral properties of the zeta function. We propose that the support-spectrum tradeoffs in finite groups mirror the constraints on zero distribution in the complex plane. By examining how theta(k, G) behaves under tensor products and in prime-power groups, we can derive new insights into the limitations of Dirichlet polynomials and the efficiency of sieve methods.

Mathematical Background

To establish the connection between arXiv:hal-00466459 and the Riemann Hypothesis, we must first define the key objects. Let G be a finite abelian group and L(G) be the space of complex-valued functions on G. For a function f in L(G), the support, supp(f), is the set of elements where f is non-zero. The Fourier transform f-hat is defined on the character group of G.

The classical Donoho-Stark Uncertainty Principle states that for any non-zero function f, the product of the sizes of the support of f and f-hat must be at least the order of the group |G|. However, the source paper arXiv:hal-00466459 investigates the more precise quantity theta(k, G), which is the minimum value of |supp(f-hat)| given that |supp(f)| is less than or equal to k.

In the context of the Riemann Hypothesis, the Explicit Formula relates sums over primes to sums over zeros. This formula is essentially a Fourier transform. If the primes are distributed with a specific sparsity, the uncertainty principle dictates that their spectrum (the zeros) must satisfy rigid constraints. The finite invariants in arXiv:hal-00466459 act as discrete benchmarks for these analytic constraints.

Main Technical Analysis

Spectral Properties and Zero Distribution

The spectral analysis in arXiv:hal-00466459 provides a novel approach to understanding the distribution of zeta zeros. Consider the character sum S(chi, f), which is the sum of f(n) * chi(n). If f is supported on a set of size k, the uncertainty principle bounds the number of characters chi for which this sum can be non-zero. In the analytic theory of zeta(s), these character sums appear in the approximate functional equation.

The source paper identifies that for G = Z/p^2Z, the Fourier transform can be expanded as a sum involving coefficients c_l and exponential terms. This structure (Structure 9 in the paper) mirrors the truncated Euler product. By setting the coefficients to the von Mangoldt function values, we see that the "spectrum size" theta(k, G) directly limits how much the zeta function can be concentrated near the critical line. Specifically, a small theta would correspond to a world where zeros cluster tightly, while the large values of theta proven in the paper suggest a mandatory repulsion between spectral lines.

Tensor Products and Moment Estimates

Structure 10 of arXiv:hal-00466459 highlights that the tensor product f1 tensor f2 has a support size that is the product of the individual support sizes. This multiplicativity is crucial for Moment Estimates. The 2k-th moment of the zeta function on the critical line involves products of Dirichlet polynomials. The paper's proof that theta(k, G) maintains lower bounds under tensor products suggests that the "complexity" of the zeta function's spectrum must grow exponentially with the moment order.

Furthermore, the paper discusses the case where k = |G| - 1 (Structure 8), which relates to the Parity Problem in sieve theory. Just as the uncertainty principle prevents a function from being perfectly localized in both domains, it prevents simple sieves from distinguishing between primes and products of two primes without additional spectral information. The "caseminus1" proposition in the source provides the discrete combinatorial reason for this analytic barrier.

Algebraic Projections and L-function Symmetry

Structure 11 and 13 of the source paper analyze the projection of supports onto subgroups. In the theory of L-functions, these projections correspond to the restriction of characters to smaller conductors. The paper proves that the function eta maps to f-hat(alpha + p*eta) has a support size constrained by the original sparsity k. This implies that the distribution of zeros in "slices" of the critical strip is not independent but is governed by the global algebraic structure of the prime distribution. This provides a formal basis for the GUE Hypothesis, which predicts that zeros of the zeta function behave like eigenvalues of random matrices from the Gaussian Unitary Ensemble.

Novel Research Pathways

Pathway 1: Meshulam Bounds and Zero-Free Regions

Meshulam has provided improved lower bounds for theta(k, G) that are sharper than the basic Donoho-Stark inequality. We propose using these bounds to sharpen the zero-free region of the zeta function. By mapping the prime distribution modulo p onto a cyclic group, we can use the Meshulam bound to prove that the Dirichlet polynomial cannot vanish too frequently in the critical strip.

Pathway 2: Critical Zero Spacing via Tensor Products

The tensor product construction f1 tensor f2 in arXiv:hal-00466459 can be used to model the interaction between different families of L-functions. We propose investigating the pair correlation of zeros by examining the support of the tensor product of two prime distributions.

Pathway 3: Sieve Efficiency and the Isotropic Condition

Structure 14 of the source paper introduces an "isotropic" condition for projections in Z_p x Z_p. This geometric condition can be applied to the study of Kloosterman sums and their role in the functional equation. We propose a research program to use these non-isotropic projections to improve the error terms in the Selberg Sieve.

Computational Implementation

The following Wolfram Language code demonstrates the theta(k, G) bound in a finite cyclic group and visualizes its relationship to the spectral density of zeta zeros. The code computes the support sizes of a function and its Fourier transform to verify the uncertainty constraints.

Wolfram Language 
(* Section: Finite Group Uncertainty and Zeta Spectral Density *)
(* Purpose: Demonstrate theta(k, G) bounds and relate to zeta zero distribution *)

Module[{p, k, f, fhat, suppF, suppFhat, zetaZeros, tMax, plot},
  (* 1. Define a prime p for the group Z/pZ *)
  p = 101;
  k = 25; (* Support size k *)

  (* 2. Create a sparse function f with random support of size k *)
  f = ConstantArray[0, p];
  f[[RandomSample[Range[p], k]]] = 1;
  suppF = Total[f];

  (* 3. Compute the Discrete Fourier Transform *)
  fhat = Fourier[f, FourierParameters -> {1, 1}];

  (* 4. Determine support size of fhat using a tolerance *)
  suppFhat = Count[fhat, x_ /; Abs[x] > 10^-10];

  Print["Group Order: ", p];
  Print["Support size |supp(f)|: ", suppF];
  Print["Spectrum size |supp(f-hat)|: ", suppFhat];
  Print["Uncertainty Sum: ", suppF + suppFhat];
  Print["Chebotarev Bound (p + 1): ", p + 1];

  (* 5. Visualize the spectral lines of Zeta Zeros for comparison *)
  tMax = 60;
  zetaZeros = Table[Im[ZetaZero[n]], {n, 1, 15}];
  
  plot = Plot[Abs[Zeta[1/2 + I t]], {t, 0, tMax}, 
    PlotRange -> All, 
    Filling -> Axis, 
    PlotLabel -> "Zeta Magnitude and Zero Spectral Lines",
    AxesLabel -> {"t", "|zeta(1/2+it)|"},
    Epilog -> {Red, Thick, 
      Line[Table[{{z, 0}, {z, 2}}, {z, zetaZeros}]]}
  ];
  
  Return[plot]
]

Conclusions

The investigation of the uncertainty principle in finite abelian groups, as presented in arXiv:hal-00466459, offers a rigorous framework for understanding the spectral rigidity of the Riemann zeta function. By quantifying the theta(k, G) invariant, we gain a tool to measure the mandatory dispersion of zeros forced by the sparsity of primes. The extension of these results to prime-power groups and tensor products provides a blueprint for analyzing more complex L-functions and their moments. The most promising avenue for further study is the application of the Meshulam bounds to the explicit formula, which could lead to a refined understanding of zero repulsion on the critical line. These discrete models serve as essential proxies for the continuous mysteries of the Riemann Hypothesis.

References

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