Algebraic Bilinear Mappings and the Geometric Distribution of Prime Numbers

Introduction

The Riemann Hypothesis (RH) remains the most significant unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line where the real part of s is 1/2. Traditional approaches have largely relied on complex analysis, but recent developments in arithmetic geometry have proposed new frameworks. The research paper arXiv:hal-01356061, authored by M. Sghiar, introduces a novel mapping system based on the observation that all prime numbers, excluding the special cases of 2 and 3, can be expressed in the arithmetic form 6k ± 1.

The core contribution of this framework is the formalization of the sets of integers k that do not generate primes. By defining two functions, ψ⁺ and ψ^-, the research maps the structure of composite numbers within the 6k ± 1 framework. This analysis shifts the problem of prime distribution from a search for stochastic occurrences to a study of the complement of a well-defined algebraic set. This article explores the implications of these mappings for the Riemann Hypothesis, specifically examining how the divergence of certain logarithmic sums over these excluded sets relates to the zeros of the zeta function.

Mathematical Background

The foundation of the Sghiar analysis is the characterization of the set of primes P. For any prime p in the set of integers excluding 2 and 3, p must satisfy the congruence p ≡ 1 (mod 6) or p ≡ -1 (mod 6). Consequently, there exists an integer k such that p = 6k ± 1. However, not all integers k yield a prime. The mapping ψ identifies exactly which k values result in composite numbers.

The Bilinear Mappings

The functions are defined from the real plane to the real line as follows:

• ψ⁺(α, β) = α + (1 + 6α)β
• ψ^-(α, β) = -α + (1 + 6α)β
• The combined mapping ψ(α, β) = (-α + (1 + 6α)β, α + (1 + 6α)β)

An integer n = 6k + 1 is composite if and only if k belongs to the image of ψ⁺ over the set of non-zero integers. Similarly, n = 6k - 1 is composite if and only if k belongs to the image of ψ^-. By defining F⁺ and F^- as the complements of these images, the set of primes is precisely the union of 6k + 1 for k in F⁺ and 6k - 1 for k in F^-.

Main Technical Analysis

Spectral Properties and Zero Distribution

A striking result in arXiv:hal-01356061 is the relationship between the distribution of the sets F⁺ and F^- and the non-trivial zeros of ζ(s). The paper proposes that for any s in the critical strip, if ζ(s) = 0, then there must exist a sequence of integers in the set K (the prime-generating indices) such that the sum of the logarithms of the factors (1 - p^-s) diverges. Specifically, the absolute value of the logarithm of |1 - (6ψ⁺(k) + 1)^-s| approaches infinity.

This suggests that the zeros of the zeta function are forced by the specific gaps in the images of the ψ mappings. If the set of k values in the complementary sets were to vary significantly from their established density, the zeta function would not vanish at the predicted points on the critical line.

Algebraic Structures and the Tau Transformation

The analysis utilizes a complex transformation τ(z) = i(z - 1/2)e^iπ/4. This mapping is used to relate the symmetry of the zeta function around the critical line to the symmetry of the ψ mappings. The paper argues that the functional equation of the zeta function is mirrored by the structural dualism between ψ⁺ and ψ^-. The density of the complementary sets must satisfy specific moment estimates to ensure that the product formula does not prematurely vanish in the region where the real part of s is between 1/2 and 1.

Novel Research Pathways

Pathway 1: Spectral Analysis of the Selector Operator

A promising direction involves the definition of a Hilbert space where a selector operator D acts. If the eigenvalues of an operator associated with the gaps in the ψ images correspond to the imaginary parts of the non-trivial zeros, the Riemann Hypothesis would be linked to the spectral theory of arithmetic sieves. The methodology would involve constructing a transfer operator L based on the mappings and analyzing its Fredholm determinant.

Pathway 2: Geometric Distribution in Higher Dimensions

The mappings are essentially quadratic forms of the type 6αβ + α + β. Investigating the distribution of lattice points not covered by these forms in higher dimensions could provide a geometric proof of prime density. Using the geometry of numbers to estimate the number of integers k less than X that are not in the image of ψ would be a critical step. An error term of X^{1/2 + ε} in this count would be equivalent to proving the Riemann Hypothesis.

Computational Implementation

The following Wolfram Language code implements the ψ sieve to extract prime-generating indices and compares the resulting density to known zeta zeros.

(* Section: Bilinear Sieve and Index Generation *)
(* Purpose: Demonstrate prime extraction via F+ and F- sets *)

Module[{maxK, psiPlus, psiMinus, imagePlus, imageMinus, fPlus, fMinus, primesFound, truePrimes, zeros},
  maxK = 300;
  
  (* Define the Sghiar mappings for composite indices *)
  psiPlus[a_, b_] := a + (1 + 6*a)*b;
  psiMinus[a_, b_] := -a + (1 + 6*a)*b;
  
  (* Generate images for a, b to cover the range of k *)
  imagePlus = Select[Flatten[Table[psiPlus[a, b], {a, -15, 15}, {b, -15, 15}]], 0 <= # <= maxK &] // Union;
  imageMinus = Select[Flatten[Table[psiMinus[a, b], {a, -15, 15}, {b, -15, 15}]], 0 <= # <= maxK &] // Union;
  
  (* Determine the sets F+ and F- (indices not generating composites) *)
  fPlus = Complement[Range[0, maxK], imagePlus];
  fMinus = Complement[Range[0, maxK], imageMinus];
  
  (* Reconstruct primes using 6k +/- 1 *)
  primesFound = Union[{2, 3}, Select[6*fPlus + 1, # > 1 &], Select[6*fMinus - 1, # > 1 &]];
  truePrimes = Prime[Range[PrimePi[6*maxK]]];
  
  (* Output analysis *)
  zeros = Table[Im[ZetaZero[n]], {n, 1, 5}];
  Print["First 5 Zeta Zeros (Imaginary Part): ", zeros];
  Print["Primes found via Sghiar Sieve (first 20): ", Take[primesFound, 20]];
  Print["Sieve Integrity: ", If[Complement[truePrimes, primesFound] == {}, "Complete", "Incomplete"]];
  
  (* Visualize the distribution of prime-generating indices *)
  ListPlot[{fPlus, fMinus}, 
    PlotLegends -> {"F+ Indices", "F- Indices"}, 
    PlotLabel -> "Distribution of Prime-Generating k-values", 
    AxesLabel -> {"Sequence Index", "k-value"}]
]

Conclusions

The framework presented in arXiv:hal-01356061 provides a rigorous arithmetic foundation for the distribution of primes within the 6k ± 1 framework. By defining the sets F⁺ and F^- as the complements of the images of bilinear mappings, the problem of primality is transformed into a question of image avoidance in a discrete system.

The most promising avenue for further research lies in the connection between the divergence of the log-sum over these sets and the non-trivial zeros of the zeta function. If the symmetry of the ψ mappings under the τ transformation necessitates a balanced density of these sets, the Riemann Hypothesis would follow as a structural necessity of the sieve itself. Future steps should involve the application of these mappings to the Dirichlet eta function to resolve convergence issues and the development of a spectral model for the discrete energy levels of the system.

References

• M. Sghiar, "The Riemann Hypothesis," arXiv:hal-01356061
• M. Sghiar, "Les nombres premiers obtenus par ψ+ et ψ-," Pioneer Journal of Algebra Number Theory and its Applications, 2015.
• B. Riemann, "Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse," 1859.