Primorial Sieve Dynamics and the Statistical Structure of the Riemann Hypothesis

Executive Summary

The research presented in arXiv:2403.17949v1 introduces a sophisticated deterministic framework for the analysis of prime distribution through recursive interval narrowing and primorial-based sieve dynamics. The core insight involves mapping prime indices to specific residue counts which characterize the density fluctuations of primes within intervals defined by primorials. This approach offers a discrete, stage-based methodology for capturing the local behavior of the prime-counting function. By demonstrating that the distribution of residues converges to a stable statistical profile as the prime index increases, the analysis provides evidence for a highly regulated spectral distribution of primes consistent with the horizontal constraints of the critical line.

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 Re(s) = 1/2. While the Prime Number Theorem provides the asymptotic density of primes, the fine-grained fluctuations in this density are what RH seeks to bound. arXiv:2403.17949v1 approaches this problem through the systematic narrowing of intervals via a multi-stage sieve process.

By utilizing primorial structures and a discrete mapping of prime gaps, the paper constructs a deterministic map of prime locations. This analysis is particularly relevant to the study of the zeta function because the zeros of ζ(s) can be viewed as the harmonics that correct the smooth distribution of primes to their actual discrete positions. This article analyzes how the convergence of residue percentages and the product-sum formulas provide a structural basis for validating the distribution of zeros on the critical line.

Mathematical Background

The fundamental unit of this analysis is the primorial, denoted by p#, which is the product of all primes less than or equal to p. The stages of the sieve represent the iterative refinement of a real interval Y. As shown in the source paper:

• Stage 2 (p=3): Narrows the interval Y to [7/6, 8/6[.
• Stage 3 (p=5): Narrows the interval Y to [37/30, 38/30[.

This recursive narrowing is a form of dynamic sieving where the surviving candidates for primality are tracked through specific counting variables. A critical constant identified in the research is defined as a = 2 * sum(x=1 to (p-1)/2) of (product(r|x, r odd prime) of (r-1)/(r-2) - 1). This constant represents the deviation of prime pair distributions from the average density, mirroring the singular series corrections found in the Hardy-Littlewood conjectures.

Main Technical Analysis

Spectral Properties and Residue Convergence

The technical core of the study is the mapping of stage indices to residue counts. Data tables provided in arXiv:2403.17949v1 show a striking trend in the vacancy rate of intervals. For s=101, the case where no primes are found in a residue class (n=0) occurs 34.52% of the time. However, by s=1000, this rate stabilizes at approximately 91.48%. This convergence suggests that primes are governed by a quasicrystal structure rather than pure randomness.

Connection to the Critical Line

The Riemann Hypothesis is equivalent to the statement that prime numbers are distributed with maximal regularity. The stability of these percentages supports this regularity. Wild fluctuations in these distributions would imply large, unpredictable clusters of primes, which would correspond to zeta zeros located off the critical line. The deterministic calculation of the constant a provides a way to approximate the sum over zeta zeros using only the prime factors of the residues, bridging the gap between discrete sieves and continuous analytic functions.

Novel Research Pathways

1. Computational Zero Detection Through Sieve Oscillations

The most promising research direction involves developing a method to detect Riemann zeta zeros through the oscillatory patterns in primorial sieve data. Each non-trivial zero contributes a periodic component to the ratio sequences. By computing the discrete Fourier transform of the normalized deviation sequence, researchers may identify spectral peaks corresponding to zeta zero ordinates.

2. Primorial Entropy and Zero Spacing

The convergence of the n=0 residue count to 91.48% suggests a saturation point in the primorial sieve. Future research should determine if the spacing between the non-trivial zeros can be mapped directly to the transitions between stages. This involves calculating the Shannon entropy of the sequence and comparing it to the statistics of zeta zero spacings.

Computational Implementation

The following Wolfram Language code implements the calculation of the structural constant and simulates the stage refinement logic discussed in the analysis.

(* Section: Primorial Sieve and Constant Analysis *)
(* Purpose: Demonstrates the calculation of the a-constant and stage refinement *)

ClearAll[primorial, distinctOddPrimeProduct, calculateA, refineStage];

primorial[k_] := Product[Prime[i], {i, 1, k}];

distinctOddPrimeProduct[x_] := Module[{factors, oddFactors},
  factors = First /@ FactorInteger[x];
  oddFactors = Select[factors, OddQ[#] && PrimeQ[#] &];
  If[Length[oddFactors] == 0, 1, 
   Product[(r - 1)/(r - 2), {r, oddFactors}]]
];

calculateA[p_] := 2 * Sum[
  (distinctOddPrimeProduct[x] - 1), 
  {x, 1, (p - 1)/2}
];

refineStage[{A_Integer, k_Integer}] := Module[
  {pk = Prime[k], window, primes},
  window = {A*pk, (A + 1)*pk};
  primes = Select[Range[window[[1]], window[[2]] - 1], PrimeQ];
  ({#, k + 1} &) /@ primes
];

(* Generate data for first 40 stages *)
data = Table[{s, calculateA[Prime[s]]}, {s, 3, 40}];
ListLinePlot[data, PlotLabel -> "Growth of Constant 'a' per Stage", 
 AxesLabel -> {"Stage s", "Constant a"}, PlotMarkers -> Automatic]

Conclusions

The analysis of the primorial-based sieve reveals a profound deterministic structure underlying the distribution of prime numbers. By shifting focus from continuous density functions to discrete residue counts, arXiv:2403.17949v1 provides a new set of tools for tackling the Riemann Hypothesis. The observed stability in the residue classes suggests a structural rigidity in prime distribution that is entirely consistent with the truth of the hypothesis. Future steps should involve formalizing the relationship between the singular series constants and the explicit formula for zeta zeros.

References

• arXiv:2403.17949v1: Riemann Hypothesis Connections and Sieve Residue Dynamics.
• Hardy, G. H., and Littlewood, J. E. (1923). On the expression of a number as a sum of primes.
• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function.