Recursive Prime Sieve Dynamics and the Geometric Distribution of Zeta Zeros

Introduction

The distribution of prime numbers is a fundamental challenge in analytic number theory, typically investigated through the zeros of the Riemann zeta function, ζ(s). The source paper, arXiv:hal-01180298, presents a novel recursive mechanism for prime generation that relies on discrete arithmetic indicators rather than transcendental approximations. This article examines how these recursive structures interface with the Riemann Hypothesis, particularly through the lens of sieve dynamics and spectral distribution.

Mathematical Background

The core construction in arXiv:hal-01180298 revolves around a primality indicator function, f(m), defined for m ≥ 4. This function uses a nested product of delta-like symbols to detect compositeness. Let δ(a, b) be an indicator function that returns 1 if a = b and 0 otherwise. The indicator is defined as the product over j and l up to a cutoff k'(m), typically the floor of the square root of m.

• Indicator Function: f(m) is the product of (1 - δ(m, jl)) for all j and l in the range [2, k'(m)].
• Selection Logic: If m is composite, there exists a pair (j, l) such that jl = m, making the product zero. If m is prime, f(m) = 1.

The paper then extends this to a formula for p_n by using a selection mechanism that counts the number of times f(m) evaluates to 1. This discrete scanning process is effectively a symbolic realization of the prime counting function, π(x).

Main Technical Analysis

Spectral Properties and Zero Distribution

The recursive system described in arXiv:hal-01180298 can be viewed as a discrete dynamical system. The "waiting time" between occurrences of f(m) = 1 represents the prime gaps. According to the explicit formula of Riemann, these gaps are influenced by the non-trivial zeros of ζ(s). If we treat the sequence f(m) as a signal, its Fourier transform would ideally exhibit peaks at frequencies corresponding to the imaginary parts of the zeta zeros on the critical line.

Sieve Bounds and Prime Density

The effectiveness of the sieve depends on the bound k'(m). In the source paper, the use of nested products ensures that no composite number escapes detection. Analytically, this relates to the Density Hypothesis. The error terms in π(x) are bounded by x^1/2 log x under the assumption of the Riemann Hypothesis. The discrete jumps in the paper's summation kernel must exactly match the fluctuations predicted by the sum over zeta zeros, implying a deep algebraic symmetry between the indicator f(m) and the zeros ρ.

Novel Research Pathways

Pathway 1: The Discrete Sieve Zeta Function

One promising direction is the construction of a "Sieve Zeta Function" based on the paper's indicator: ζ_sieve(s) = ∑ f(m) m^-s. This is equivalent to the Prime Zeta Function, P(s). Since P(s) has singularities at s = ρ/k, where ρ are the zeros of ζ(s), the nested products in arXiv:hal-01180298 provide a direct algebraic link to the singular points of the zeta function. Researching the convergence of this series could yield new bounds on the real parts of the zeros.

Pathway 2: Spectral Analysis of Recursion Operators

Another pathway involves defining a linear operator T that maps the sequence of the first n-1 primes to p_n. By analyzing the eigenvalue distribution of this operator, one might find correlations with the Montgomery-Odlyzko law for zeta zero spacing. This would bridge the gap between constructive arithmetic and the statistical properties of L-functions.

Computational Implementation

(* Section: Recursive Prime Indicator and Zeta-Zero Correlation *)
(* Purpose: Implement the hal-01180298 sieve and compare with Zeta zeros *)

ClearAll[sieveIndicator, recursivePi, rhBound];

(* Define the indicator f(m) from the paper using sqrt cutoff *)
sieveIndicator[m_Integer] := Module[{kLimit},
  If[m < 2, Return[0]];
  kLimit = Floor[Sqrt[m]];
  Product[
    Product[If[m == j*l, 0, 1], {l, 2, kLimit}],
    {j, 2, kLimit}
  ]
];

(* Cumulative sum of the indicator to represent pi(x) *)
recursivePi[x_Integer] := Total[Table[sieveIndicator[m], {m, 2, x}]];

(* Riemann Hypothesis error bound: sqrt(x) log(x) / (8 pi) *)
rhBound[x_] := (1/(8 * Pi)) * Sqrt[x] * Log[x];

(* Generate comparison data for x up to 200 *)
data = Table[{
  x,
  recursivePi[x],
  LogIntegral[x] // N,
  Abs[recursivePi[x] - LogIntegral[x]] // N,
  rhBound[x] // N
}, {x, 10, 200, 10}];

(* Plotting the error against the RH bound *)
Plot[{
  Abs[recursivePi[Floor[t]] - LogIntegral[t]],
  rhBound[t]
}, {t, 2, 300},
  PlotStyle -> {Blue, Red},
  PlotLegends -> {"Actual Error", "RH Bound"},
  AxesLabel -> {"x", "Magnitude"},
  PlotLabel -> "Sieve Error vs Riemann Hypothesis Bounds"]

Conclusions

The recursive formulas in arXiv:hal-01180298 provide a rigid algebraic framework for prime generation. While computationally intensive, the structure of the nested products offers a unique perspective on how primality is encoded in discrete sequences. The most significant avenue for future research lies in the spectral analysis of the sieve indicator, which may eventually demonstrate that the convergence of such arithmetic formulas necessitates the truth of the Riemann Hypothesis. By linking the discrete jumps of the prime counting function to the analytic properties of ζ(s), we move closer to a unified theory of prime distribution.

References

• arXiv:hal-01180298: A formula for the nth prime number.
• Riemann, B. (1859). "On the Number of Primes Less Than a Given Magnitude."
• arXiv:1004.1563: Asymptotic densities and the Mobius function.