The Symmetry of Primal Radii: Connecting Parametric Prime Decompositions to the Critical Line

Introduction

The distribution of prime numbers remains one of the most profound mysteries in mathematics, sitting at the intersection of additive number theory and complex analysis. While the Prime Number Theorem provides a global view of prime density, the fine-grained local behavior—specifically how primes are distributed symmetrically around a given integer—is the central concern of the Strong Goldbach Conjecture and, by extension, the Riemann Hypothesis. The research paper arXiv:hal-00677734 introduces a rigorous algebraic framework centered on the concept of the primal radius, suggesting that for every integer x greater than or equal to 3, there exists a radius r such that x - r and x + r are both prime.

This analysis explores the mathematical structures presented in the source, specifically the system of equations involving prime shifts and algebraic constants alpha and beta. We aim to demonstrate how the existence of the primal radius is not merely an additive curiosity but is deeply constrained by the analytic properties of the Riemann zeta function, zeta(s). The Riemann Hypothesis asserts that all non-trivial zeros of zeta(s) lie on the critical line Re(s) = 1/2. If this symmetry in the complex plane holds, it implies a corresponding symmetry in the distribution of primes, providing the necessary density to guarantee the existence of the primal radius for all x.

The contribution of this article is to bridge the gap between the algebraic forcing methods found in the source paper and the spectral distribution of prime numbers. By examining the constants k, m, alpha, and beta defined in the source, we can formulate a new perspective on how the error terms in prime distribution—governed by the zeros of zeta(s)—must behave to satisfy the algebraic identities derived in arXiv:hal-00677734.

Mathematical Background

The foundational framework of arXiv:hal-00677734 begins with the parametric representation of four variables related to prime pairs. For given primes p₁ and p₂, the paper defines a system where x₁ = p₁ + 2b, x₂ = p₂ - 2b, x₃ = p₂ + 2b, and x₄ = p₁ - 2b. Here, b represents a displacement parameter that characterizes the geometric relationship between the original primes and their transformed counterparts. This parametric system establishes a rich algebraic structure that constrains the possible relationships between these variables.

The paper derives expressions for an integer x and a coordinate y: x = (p₁ + p₂)/2 + b and y = (p₁ - p₂)/2 + b. The core of the argument rests on the Theorem of Primal Radius, which states that for all x greater than or equal to 3, there exist p₁ and p₂ such that 2x = p₁ + p₂. In the language of the source paper, this is equivalent to proving that the parameter b must vanish (b = 0) under certain algebraic constraints. If b = 0, then x is precisely the midpoint of the two primes, and r = (p₁ - p₂)/2 becomes the primal radius.

Connecting this to the Riemann zeta function requires the von Mangoldt function, Lambda(n), which weights prime powers. The Riemann Hypothesis is equivalent to the statement that the summatory function of Lambda(n), denoted as psi(x), satisfies psi(x) = x + O(x^1/2 log^2 x). The error term x^1/2 is critical; it represents the square-root cancellation expected from a random-like distribution of primes, which is necessary to ensure that symmetric prime pairs exist frequently enough to satisfy the primal radius theorem.

Main Technical Analysis

Spectral Properties and Zero Distribution

The distribution of zeros of the zeta function is a critical aspect of the Riemann Hypothesis. The source paper's equations can be analyzed in the context of spectral properties, potentially revealing insights into the zero distribution. The relationship between x₁, x₂, and the primes p₁, p₂ is connected to the spectral properties of certain operators related to the zeta function. Specifically, the paper introduces ratio parameters k and k' defined as k = 2b/(p₁ - p₂) and k' = -2b/(x₁ - x₂).

A crucial development in the analysis occurs when the paper establishes relationships between these ratio parameters through scaling factors alpha and beta. The condition k = alpha k' leads to the fundamental constraint x₁ - x₂ = -alpha(p₁ - p₂). This relationship forces the displacement parameter to satisfy b = -(alpha + 1)(p₁ - p₂)/4. The critical observation is that consistency conditions require alpha = beta = -1, which immediately implies b = 0.

Algebraic Constraints and the Vanishing of b

The vanishing of the displacement parameter represents a critical point in the parametric system where the geometric relationships simplify dramatically. In the context of the Riemann Hypothesis, these critical points bear striking resemblance to the critical line where the zeta function exhibits its most profound properties. Just as the critical line represents the optimal location for zeta zeros in terms of balancing growth rates, the condition b = 0 represents an optimal configuration for prime decompositions that minimizes geometric displacement.

Furthermore, the emergence of the golden ratio phi in the expressions x = (phi / (phi - 1)) p₂ and y = (1 / (phi - 1)) p₂ is particularly significant. This appearance of phi suggests underlying harmonic relationships in the prime distribution that parallel the quasi-periodic behavior observed in the distribution of zeta zeros. The golden ratio's connection to continued fractions and optimal approximation properties provides a bridge between the discrete prime distribution and the continuous properties of the zeta function.

Novel Research Pathways

The connections established between the parametric framework of arXiv:hal-00677734 and zeta function theory suggest several promising research directions.

• Parametric L-functions and Extended Zeta Relations: One could construct L-functions based on the parametric relationships identified in the source paper. By defining a parametric zeta function that sums over prime pairs satisfying the constraints, we can study how the critical condition alpha = -1 affects the analytic properties of the associated function. We hypothesize that b = 0 manifests as special zeros or poles that align with the critical line.
• Spectral Analysis of Prime Gap Oscillations: The oscillatory behavior implied by the golden ratio relationships suggests that prime gaps exhibit patterns that could be analyzed using techniques from spectral theory. Studying the correlation function of the displacement parameter b across sequences of primes may reveal characteristic frequencies related to the imaginary parts of zeta function zeros.
• Geometric Moduli Spaces: We propose constructing a moduli space whose points represent equivalence classes of prime quadruples satisfying the parametric relationships. The critical locus where b = 0 would define a subvariety encoding optimal prime configurations. The geometric properties of this locus, such as its dimension and singularity structure, could provide insights into universal properties of prime distributions.

Computational Implementation

The following Wolfram Language implementation investigates the "primal radius" viewpoint by searching for the minimal radius r for a range of integers and visualizing the imaginary parts of zeta zeros to demonstrate the underlying spectral landscape.

(* Section: Primal Radius Search and Zeta Zero Spectrum *)
(* Purpose: Demonstrate the primal radius r (where b=0) and zeta zeros *)

Module[
  {xMax = 400, rLimit = 250, results, zeros, radiusPlot, zeroPlot},

  (* Find smallest r such that x-r and x+r are prime *)
  results = Table[
    {x, Catch[
      Do[
        If[PrimeQ[x - r] && PrimeQ[x + r], Throw[r]],
        {r, 0, rLimit}
      ];
      Missing["NotFound"]
    ]},
    {x, 3, xMax}
  ];

  (* Calculate ordinates of the first 25 zeta zeros *)
  zeros = Table[Im[ZetaZero[n]], {n, 1, 25}];

  (* Plotting the primal radius distribution *)
  radiusPlot = ListPlot[
    DeleteMissing[results],
    PlotLabel -> "Minimal Primal Radius r for Integer x",
    AxesLabel -> {"x", "r"},
    Filling -> Axis,
    PlotStyle -> Blue
  ];

  (* Plotting imaginary parts of zeta zeros *)
  zeroPlot = ListPlot[
    zeros,
    PlotLabel -> "Imaginary Parts of First 25 Zeta Zeros",
    AxesLabel -> {"n", "Im(rho)"},
    Filling -> Axis,
    PlotStyle -> Red
  ];

  Print[radiusPlot];
  Print[zeroPlot];
  Print["Average r found: ", Mean[DeleteMissing[results[[All, 2]]]]];
]

Conclusions

The analysis of the primal radius framework reveals profound connections between algebraic prime decomposition and the fundamental structures underlying the Riemann Hypothesis. The emergence of critical conditions where the displacement parameter b vanishes provides a discrete analog to the critical line phenomena in zeta function theory. This suggests that the distribution of primes is governed by optimization principles that align with the spectral properties of the zeta function. The most promising avenue for future research lies in the spectral decomposition of prime gap patterns, which may provide new tools for studying zeta zero distributions and potentially lead to improved bounds on zero-free regions.

References

• arXiv:hal-00677734
• Edwards, H.M. (1974). Riemann's Zeta Function. Academic Press.
• Titchmarsh, E.C. (1986). The Theory of the Riemann Zeta Function. Oxford University Press.
• Montgomery, H. L. (1973). "The pair correlation of zeros of the zeta function."