Harmonic Analysis of Prime Progressions: A Spectral Bridge to the Riemann Hypothesis

Executive Summary

This article explores the deep structural relationship between the additive combinatorics of three-term arithmetic progressions (3-APs) and the analytic properties of the Riemann zeta function. Drawing from the Fourier-analytic framework presented in arXiv:cel-00963631v2_document, we analyze the trilinear form Λ₃ and its spectral representation. The key insight is that the "pseudorandomness" required to guarantee the existence of 3-APs in sparse sets like the primes is fundamentally constrained by the distribution of the non-trivial zeros of ζ(s). By linking the decay of Fourier coefficients to the explicit formula for the von Mangoldt function, we demonstrate that the Riemann Hypothesis (RH) provides the optimal bound for the error terms in progression counts. This connection offers a promising pathway for translating combinatorial density increments into new constraints on the critical line.

Introduction

Roth’s Theorem (1953) stands as a cornerstone of additive combinatorics, asserting that any subset of integers with positive density contains infinitely many 3-term arithmetic progressions. The proof strategy, particularly the Fourier-analytic approach detailed in arXiv:cel-00963631v2_document, relies on a dichotomy between structure and randomness. If a set lacks the expected number of progressions, it must exhibit a large Fourier coefficient, indicating a correlation with an arithmetic progression of even greater density.

While Roth's original theorem applied to dense sets, extending these results to sparse sequences like the primes requires the "transference principle." This principle involves majorizing the primes with a pseudorandom weight function. The effectiveness of this majorant is determined by the cancellation of exponential sums over the primes, a problem essentially equivalent to understanding the distribution of prime numbers. Consequently, the analytic behavior of the Riemann zeta function ζ(s) becomes the governing force behind the additive patterns found in the primes.

Mathematical Background

The central object of study in arXiv:cel-00963631v2_document is the counting operator for 3-term arithmetic progressions in the cyclic group Z_N. For functions f₁, f₂, and f₃, the trilinear form is defined as:

Λ₃(f₁, f₂, f₃) = (1/N²) ∑_{n1 + n3 = 2n2} f₁(n₁) f₂(n₂) f₃(n₃)

Using the Discrete Fourier Transform (DFT), where f_hat(k) = (1/N) ∑ f(n) exp(-2πikn/N), the source paper establishes the fundamental spectral identity:

Λ₃(f₁, f₂, f₃) = ∑_{k in ZN} f_hat₁(k) f_hat₂(-2k) f_hat₃(k)

In this representation, the k=0 term captures the main density contribution, while the nonzero terms represent spectral fluctuations. When applied to the primes, the functions f_i are replaced by the von Mangoldt function Λ(n). The Fourier coefficients f_hat(k) then become exponential sums whose magnitudes are determined by the location of the non-trivial zeros ρ = σ + iγ of the zeta function.

Main Technical Analysis

Spectral Properties and Zero Distribution

The connection to the Riemann Hypothesis is most visible in the decay of the Fourier coefficients f_hat(k). For prime-like sequences, the explicit formula relates these sums to the zeros of ζ(s). Specifically, the exponential sum S(α) = ∑ Λ(n) exp(2πinα) can be approximated by a main term minus a sum over zeta zeros: ∑ x^ρ/ρ. Under the Riemann Hypothesis, where all σ = 1/2, these sums exhibit near square-root cancellation, roughly of the order N^1/2.

In the framework of arXiv:cel-00963631v2_document, this cancellation ensures that the nonzero Fourier coefficients are small. If the Riemann Hypothesis were false, a zero with σ > 1/2 would create a "spectral spike" at certain frequencies, potentially disrupting the pseudorandomness required for the transference principle. Thus, RH provides the theoretical minimum for the "noise" in the distribution of primes across arithmetic progressions.

Sieve Bounds and Prime Density

The source paper utilizes a pseudorandom measure ν to majorize the function of interest. For the primes, this measure is often constructed via sieve methods. The paper provides an inequality for the smoothed function f₁:

|f₁(n)| ≤ ∑_{r in ZN} |ν_hat(r)| |β_hat(r)|²

where β is a Bohr set cutoff. The quality of this bound depends on the Fourier decay of ν. If RH holds, the measure ν satisfies the Gowers uniformity conditions more strongly, allowing for a more efficient density increment. This implies that the threshold for finding 3-APs in subsets of primes is directly linked to the vertical distribution of zeta zeros. A tighter clustering of zeros would correspond to a more "structured" distribution of primes, while the GUE (Gaussian Unitary Ensemble) spacing predicted by RH corresponds to maximum pseudorandomness.

Moment Estimates and Growth Rates

The L^p norms of Fourier coefficients discussed in arXiv:cel-00963631v2_document are discrete analogues of the moments of the zeta function on the critical line. The Lindelof Hypothesis, which follows from RH, suggests that the zeta function grows slower than any power of t. In the context of 3-APs, this bound implies that the sum of the cubes of the Fourier coefficients (the spectral bias) is dominated by the major arcs, with the minor arcs contributing only negligible error. This stability is what allows the density increment argument to terminate in a finite number of steps, providing a lower bound on the number of progressions.

Novel Research Pathways

1. The Zeta-Gowers Correlation

We propose investigating a "Zeta-Gowers Norm" that weights the Fourier coefficients of a set by their proximity to the imaginary parts of zeta zeros. This research would determine if sets that avoid 3-term progressions must necessarily have a spectral signature that mimics the distribution of ζ(s) zeros. The methodology involves using the density increment strategy from the source paper to see if the "large Fourier coefficient" found in the structured case must align with a resonance of the zeta function.

2. Arithmetic Progression Zeta Functions

A new class of Dirichlet series can be defined where the coefficients are the Λ₃ counts for a given density α over increasing intervals N. The analytic properties of this "AP-Zeta Function," such as its pole structure and zero-free regions, would encode the combinatorial difficulty of finding progressions in sparse sets. This would bridge the gap between the Hardy-Littlewood circle method and the Green-Tao theorem.

3. Spectral Gaps and the Critical Strip

By treating the imaginary parts of zeta zeros themselves as a point process, one can apply the 3-AP counting operator to the zeros. If the zeros are "repelled" as suggested by Montgomery’s Pair Correlation Conjecture, they should contain fewer 3-APs than a random Poisson process. Proving a "Soft Roth Theorem" for the zeros of ζ(s) could provide a new statistical test for the Riemann Hypothesis.

Computational Implementation

(* Section: Spectral Analysis of 3-AP Fourier Coefficients *)
(* Purpose: Demonstrate the Fourier identity for Lambda_3 and its spectral properties *)

Module[{N = 256, f, fhat, lambda3Fourier, lambda3Direct, k, n, r, specPlot},
  (* Define a weight function on Z_N (normalized indicator of primes) *)
  f = Table[If[PrimeQ[n], Log[n], 0.0], {n, 1, N}];
  f = f / Mean[Select[f, # > 0 &]];

  (* Discrete Fourier Transform following arXiv:cel-00963631v2_document *)
  fhat = Table[(1/N) * Sum[f[[n]] * Exp[-2 * Pi * I * (k - 1) * (n - 1) / N], {n, 1, N}], {k, 1, N}];

  (* Lambda_3 via the Fourier identity: Sum of fhat(k)^2 * fhat(-2k) *)
  lambda3Fourier = Re[Sum[fhat[[k]]^2 * fhat[[Mod[-2*(k-1), N] + 1]], {k, 1, N}]];

  (* Direct calculation of Lambda_3 counting triples n, n+r, n+2r *)
  lambda3Direct = (1/N^2) * Sum[f[[n+1]] * f[[Mod[n+r, N]+1]] * f[[Mod[n+2*r, N]+1]], {n, 0, N-1}, {r, 0, N-1}];

  Print["Lambda_3 (Fourier spectral sum): ", lambda3Fourier];
  Print["Lambda_3 (Direct combinatorial sum): ", lambda3Direct];

  (* Plotting the spectrum magnitude to visualize major and minor arcs *)
  specPlot = ListLinePlot[Abs[fhat], 
    PlotLabel -> "Fourier Spectrum Magnitude of Prime Weights", 
    AxesLabel -> {"Frequency k", "|f_hat(k)|"}, 
    PlotRange -> All];
  Print[specPlot];
]

Conclusions

The Fourier-analytic framework for Roth’s Theorem provided by arXiv:cel-00963631v2_document offers a powerful lens for examining the distribution of primes. By decomposing the 3-AP counting functional into its spectral components, we see that the existence of additive patterns is inextricably linked to the decay of exponential sums. The Riemann Hypothesis, by enforcing optimal cancellation in these sums, ensures that the primes behave with the maximum possible pseudorandomness. Future research into the Gowers norms of the zeta function and the construction of AP-specific zeta functions may provide the necessary tools to finally bridge the gap between additive combinatorics and the zeros of the zeta function on the critical line.

References

• arXiv:cel-00963631v2_document: Roth's Theorem and Fourier Analysis on Finite Groups.
• Roth, K. F. (1953). On certain sets of integers. Journal of the London Mathematical Society.
• Green, B., & Tao, T. (2008). The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics.
• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function.