The Geometry of Primes: Mapping the 6n+/-1 Selector onto the Critical Line

Executive Summary

The source paper arXiv:1709.02439v1 introduces a deterministic framework for identifying prime numbers and twin prime pairs through a prime selector function defined as k(a, b) = 2ab + a + b. By mapping all odd numbers m = 2k + 1 to integer indices k, the research demonstrates that an odd number is composite if and only if its index can be represented by this specific bilinear form. This recontextualization shifts the study of primality from elusive analytic distributions to a structured, multidimensional sieve of composite order numbers.

The main connection to the Riemann Hypothesis (RH) lies in the regularity and dispersion of the values skipped by this selector function. RH is fundamentally a statement about the error term in the distribution of primes, which corresponds to the oscillations of the zeta function ζ(s) along the critical line. By analyzing the collisions and gaps within the k(a, b) series, we can construct a combinatorial bridge to the zeta zeros. This approach is promising because it transforms the analytic problem of zero-free regions into a discrete problem of lattice point distribution and indicator function cancellation.

Introduction

The distribution of prime numbers is governed by the zeros of the Riemann zeta function, yet the bridge between the discrete nature of primes and the continuous nature of complex analysis remains one of the greatest challenges in mathematics. The paper arXiv:1709.02439v1 provides a novel lens by focusing on the 6n ± 1 architecture. Since every prime greater than 3 must reside in one of these two residue classes, the primality of 6n - 1 and 6n + 1 can be analyzed through the integer index n.

This article synthesizes the selector function framework with analytic number theory to explore how the density of composite numbers—as generated by the k(a, b) function—dictates the behavior of the prime-counting function π(x). We examine the spectral properties of these selector "hits" and propose research pathways that link the fluctuations in composite density to the imaginary parts of the non-trivial zeros of ζ(s).

Mathematical Background

The foundation of this framework is the Prime Selector Function. For any odd number m = 2k + 1, the number is composite if it can be factored as (2a + 1)(2b + 1) for positive integers a and b. Expanding this product yields 4ab + 2a + 2b + 1, which simplifies to the identity:

k(a, b) = 2ab + a + b

If a given index k cannot be expressed in this form, the corresponding odd number 2k + 1 is prime. The paper arXiv:1709.02439v1 extends this to the 6n ± 1 system by defining indicator variables K_- and K₊. These variables count the number of representations of n that lead to composite values in the 6n - 1 and 6n + 1 sequences respectively. A twin prime pair is identified precisely when K_- + K₊ = 0.

In the context of the Riemann zeta function, ζ(s) = ∑ n^-s, the primes are the fundamental building blocks of the Euler product. The Riemann Hypothesis asserts that the fluctuations of the prime-counting function are bounded by x^1/2 log x. In the selector framework, these fluctuations are represented by the "gaps" between values produced by k(a, b). The density of these gaps is intrinsically tied to the distribution of the zeros of ζ(s) on the critical line Re(s) = 1/2.

Main Technical Analysis

Spectral Properties and Zero Distribution

The selector function k(a, b) can be viewed as a mapping that populates the set of composite indices. To connect this to RH, we consider the generating function F(z) = ∑ z^{k(a, b)}. The coefficients of this power series reflect the number of ways an integer can be represented as a composite index. The behavior of this function near the unit circle is related to the distribution of primes via the circle method and the properties of L-functions.

If the Riemann Hypothesis holds, the distribution of primes should exhibit a level of quasi-randomness similar to the eigenvalues of random matrices. In the 6n ± 1 framework, this translates to a requirement that the indicators K_- and K₊ do not cluster in a way that violates the square-root cancellation principle. The spectral density of the sequence S(n) = K_-(n) + K₊(n) should contain frequency components that correspond to the imaginary parts of the zeta zeros.

Sieve Bounds and Prime Density

The 6n ± 1 architecture allows for a more refined sieve than the classical Sieve of Eratosthenes. By treating n as the primary variable, the "order" of primes—as described in arXiv:1709.02439v1—is determined by the number of factors. A first-order prime has a unique representation in the system, while higher-order composites are hit multiple times by the selector function.

Analytically, the explicit formula relates the sum of the von Mangoldt function to the zeros of the zeta function. The K indicators act as discrete proxies for these fluctuations. The goal is to prove that the sum of these indicators over a range N behaves with the regularity required by RH. This involves bounding the number of lattice points on the hyperbolas defined by the k(a, b) equations.

Novel Research Pathways

Pathway 1: The Dirichlet Selector Transform

One promising direction is to define a Dirichlet series based on the selector hits. Let r(m) be the number of ways an odd number m is hit by the selector. We can analyze the series L(s) = ∑ r(m) m^-s. This series is naturally related to ζ(s)². By studying the analytic continuation of L(s) and its poles, we can extract information about the distribution of primes. The methodology involves using Mellin transforms to relate the selector's average behavior to the critical line.

Pathway 2: Harmonic Analysis of Twin Prime Gaps

A second pathway involves the Fourier analysis of the gaps between indices where K_- + K₊ = 0. If we treat the sequence of twin prime indices as a signal, its power spectrum should exhibit peaks at frequencies related to the zeta zeros. This research would use the 6n ± 1 framework to generate large datasets of twin prime indices and apply high-resolution spectral estimators to verify the correlation with the imaginary parts of ζ(s) zeros.

Computational Implementation

The following Wolfram Language code implements the 6n ± 1 selector logic to visualize the density of composite hits and compare the discrete indicator function with the oscillations of the Riemann zeta function along the critical line.

(* Section: 6n +/- 1 Sieve and Selector Analysis *)
(* Purpose: Visualize the composite indicator K and compare with Zeta Z-function *)

Module[{maxN = 500, selectorHits, kSum, zetaFluctuations, tMax = 60},
  (* Generate composite indices from the k(a, b) = 2ab + a + b identity *)
  selectorHits = Union[Flatten[Table[2*a*b + a + b, {a, 1, 50}, {b, 1, 50}]]];
  selectorHits = Select[selectorHits, # <= maxN &];

  (* Create the indicator sequence: 1 if composite, 0 if prime candidate *)
  kSum = Table[If[MemberQ[selectorHits, n], 1, 0], {n, 1, maxN}];

  (* Calculate Zeta fluctuations for comparison *)
  zetaFluctuations = Table[RiemannZ[t], {t, 0, tMax, 0.2}];

  (* Output visual comparison *)
  Print["Twin Prime Index Candidates Found: ", 
    Count[Table[If[MemberQ[selectorHits, n], 1, 0], {n, 1, maxN}], 0]];

  GraphicsColumn[{
    ListPlot[kSum, Filling -> Axis, 
      PlotLabel -> "Composite Selector Indicator (6n +/- 1 Context)", 
      AxesLabel -> {"n", "Hit"}, PlotStyle -> Red, ImageSize -> Large],
    ListLinePlot[zetaFluctuations, 
      PlotLabel -> "Riemann Z-Function on the Critical Line", 
      AxesLabel -> {"t", "Z(t)"}, PlotStyle -> Blue, ImageSize -> Large]
  }]
]

Summary of key findings: The 6n ± 1 framework provides a deterministic map of the prime landscape. The most promising avenue for further research is the spectral analysis of the selector function's generating function, which could link the dispersion of composite indices directly to the zeros of ζ(s). Specific next steps include a rigorous bound on the growth of the K indicators and a cross-correlation study between the k(a, b) lattice and the known zeros of the Riemann zeta function.

References

• arXiv:1709.02439v1: The Prime Number Selector Function 2ab + a + b.
• arXiv:1708.07884v1: Evidence for the existence of infinite Twin Prime Pairs.
• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function.