Multi-Dimensional Tensor Sieve Dynamics and the Critical Line of Prime Distribution

Introduction

The distribution of prime numbers is a central mystery of number theory, traditionally analyzed through the analytical lens of the Riemann zeta function. However, the source paper arXiv:hal-02549967v15 introduces a distinct paradigm: the representation of natural numbers as an infinite-dimensional tensor. By constructing a Half Infinite Tensor (MN) through the outer product of prime-power vectors, the paper provides a multi-linear algebraic framework for understanding prime density and sieve dynamics.

The significance of this approach lies in its shift from the continuous frequency domain of the zeta function to a discrete, multi-linear algebraic structure. By analyzing how systematic thread removals—representing the elimination of multiples of a specific prime—collapse the tensor, we gain a geometric realization of the fundamental theorem of arithmetic. This article synthesizes these concepts to explore connections with the Riemann zeta function and the distribution of zeros on the critical line.

Mathematical Background

The foundational structure introduced in arXiv:hal-02549967v15 begins with the representation of natural numbers through infinite-dimensional prime-based vectors. For each prime p, we define a vector p = [p^0, p^1, p^2, ...]. The MN-tensor is then constructed as the outer product: MN = 2 x 3 x 5 x 7 x ... .

In this framework, every natural number occurs exactly once as a tensor entry. The paper identifies two specific sieving processes that operate on this structure:

• Mersenne Sieve (MeS): A process that removes numbers based on threads defined by prime indices. For example, step 2 removes 1/3 of the remaining numbers, leaving a fraction of 2/3.
• Inverse Sundaram Sieve (SuS-1): A modified approach where the reduction ratios differ, such as removing 1/4 of the remaining numbers in a given step, leaving 3/4.

These ratios (2/3, 4/5, 6/7, etc.) represent the local density of integers co-prime to a primorial. In analytic number theory, the product of these ratios is governed by Mertens' theorems, which connect directly to the Euler product of the zeta function.

Main Technical Analysis

Spectral Properties and Tensor Collapse

The representation of natural numbers as an outer product provides a discrete mapping of the Euler product formula. The Riemann zeta function, zeta(s), can be viewed as the contraction of the MN-tensor with a weight tensor where each element is defined by the value n^-s. Because the MN-tensor is constructed via outer products, the contraction decomposes into a product of independent sums, mirroring the Euler product: Product (1 - p^-s)^-1.

The threads discussed in the paper represent the individual dimensions of this infinite-dimensional space. Removing a thread is effectively setting one of the factors in the Euler product to unity, thereby sieving the zeta function itself. The Riemann Hypothesis (RH) asserts that the error terms in prime distribution are as small as possible. In the tensor model, this corresponds to the maximum possible uniformity in the collapse of tensor dimensions.

Twin Prime Heuristics and Nested Integrals

The paper proposes a novel formula for the number of twin primes, pi_G2(n), involving a nested logarithmic integral: pi_G2(n) is approximately proportional to the integral of dx / (ln(x) ln(li(x))), where li(x) is the logarithmic integral. This suggests a second-order density effect that warrants comparison with the standard Hardy-Littlewood conjectures.

Analytically, if we assume the Riemann Hypothesis, the distribution of primes is as random as possible given the constraints of the sieve. The source paper's use of 15n + k progressions demonstrates how density is distributed across reduced residue classes. The RH extends this by asserting that the error in each of these classes is essentially symmetric, preserving the balance of the MN-tensor slices.

Novel Research Pathways

1. Spectral Analysis of Thread-Removal Operators

A promising direction is to define a linear operator T acting on the MN-tensor such that thread removal corresponds to a projection onto a subspace. Investigating whether these operators can be represented on Hilbert spaces may reveal spectral behavior controlled by the zeros of the zeta function.

• Formulation: Define the operator trace in terms of the retention rates (p-1)/p.
• Connection: Relate the eigenvalue distribution of the composite projection operator to the spacing of zeta zeros.

2. Moment Methods and Sieve Efficiency

The double-logarithmic structure observed in the twin prime counting formula suggests an approach based on moment methods. By studying the k-th moment of the sieve-corrected counting function, researchers might find new bounds on the zero-free region of the zeta function.

• Formulation: Analyze the asymptotic growth of moments for the difference between the tensor-sieve approximation and the true count.
• Methodology: Use the explicit formula to express the remainder in terms of sums over zeta zeros.

Computational Implementation

(* Section: Tensor Sieve Density and Twin Prime Estimation *)
(* Purpose: Simulate thread removal and calculate twin prime heuristics *)

Module[{nMax = 10000, sieveRatios, twinIntegral, xVals, liApprox},
  (* Calculate (p-1)/p ratios for the first 20 primes *)
  sieveRatios = Table[{Prime[i], (Prime[i] - 1.0)/Prime[i]}, {i, 1, 20}];
  
  (* Define the nested integral heuristic for twin primes from hal-02549967v15 *)
  twinIntegral[n_] := NIntegrate[1/(Log[x] * Log[LogIntegral[x]]), {x, 2, n}];
  
  (* Output the cumulative retention product (Euler product proxy) *)
  Print["Cumulative Retention Factors:", FoldList[Times, 1, sieveRatios[[All, 2]]]];
  
  (* Compare heuristic values at different scales *)
  xVals = {100, 1000, 10000};
  Print["Twin Prime Heuristic values:", Table[{x, twinIntegral[x]}, {x, xVals}]];
  
  (* Visualize the collapse of the tensor density via thread removal *)
  ListStepPlot[sieveRatios, 
    PlotLabel -> "MN-Tensor Dimension Collapse", 
    AxesLabel -> {"Prime Index", "Retention Ratio"}, 
    Filling -> Bottom]
]

Conclusions

The analysis of arXiv:hal-02549967v15 reveals a robust structural approach to prime distribution. By framing the integers as an infinite-dimensional tensor, the paper provides a visual and algebraic language for the sieving processes that underlie the Euler product. The most promising avenue for further research is the spectral analysis of the MN-tensor operators. If the threads can be mapped to the operators of a quantum system, the Riemann Hypothesis would correspond to the requirement that the system energy levels are stable under the removal of prime dimensions. Next steps include formalizing the error budget of the nested logarithmic integral against standard sieve upper bounds.

References

• arXiv:hal-02549967v15
• Hardy, G. H., and Littlewood, J. E. (1923). Some problems of Partitio Numerorum; III. Acta Mathematica.
• Montgomery, H. L., and Vaughan, R. C. (2007). Multiplicative Number Theory I: Classical Theory.