Recursive Sieve Identities and the Fine Structure of Prime Distribution near the Critical Line

Executive Summary

The research presented in arXiv:1406.0397v2 develops an explicit recursive sieve framework based on inclusion-exclusion principles to count primes and twin primes within short intervals, specifically the interval between consecutive squares. The central insight is the decomposition of these counts into alternating sums of floor functions, such as [x/p], which allows for a deterministic rather than purely probabilistic analysis of prime density. The connection to the Riemann Hypothesis is structural: the error terms in these sieve identities are governed by the fluctuations of the Mobius function and the distribution of the zeros of the Riemann zeta function. This approach is promising because it translates the analytic problem of zero-free regions into a discrete discrepancy problem involving the fractional parts of floor functions, offering a concrete pathway to bounding prime density fluctuations at the square-root scale.

Introduction

The distribution of prime numbers is fundamentally linked to the behavior of the Riemann zeta function, ζ(s). While the Prime Number Theorem provides a global average, the local distribution in short intervals remains a subject of intense study. The paper arXiv:1406.0397v2 addresses this by examining the interval (n², (n+1)²], which has a length of approximately 2n. This scale is particularly significant because it coincides with the error term magnitude predicted by the Riemann Hypothesis (RH).

The contribution of this analysis is the formalization of a recursive sieve, denoted as σ(η_n), which partitions the integers in a short interval by their least prime factor. By using exact floor-function counts, the author avoids the initial need for analytic approximations, creating a combinatorial bridge to the explicit formula. We explore how these structures relate to the critical line Re(s) = 1/2 and how the convergence of these recursive bounds provides evidence for the regular distribution of primes.

Mathematical Background

The foundation of the source paper is a Legendre-type sieve applied to the set of integers η_n. A composite integer m in the interval (n², (n+1)²] must have a prime factor p less than or equal to n+1. Consequently, sieving out multiples of primes up to n+1 in this interval is sufficient to identify all primes within it. The key mathematical object is the inclusion-exclusion sum involving products of primes, C_j,i-1,k, which represents a product of j distinct primes chosen from the first i-1 primes.

The counting function σ(η_n) is expressed as a sum of terms like [(n+1)²/d] - [n²/d]. Replacing these floors with their real-valued arguments leads to a main term involving the partial Euler product: product (1 - 1/p). This connects directly to the reciprocal of the zeta function, as 1/ζ(s) is the generating function for the Mobius function μ(n). The paper also extends this logic to twin primes, π₂(η_n), using products of the form (1 - 2/p), which are relevant to the Hardy-Littlewood constants and the zeros of Dirichlet L-functions.

Sieve Bounds and Prime Density

The technical analysis in arXiv:1406.0397v2 focuses on bounding the discrepancy between the exact count and the density-based approximation. The identity for σ(η_n) involves nested alternating sums that can be simplified into the form: (2n+1) sum (1/p_i) product (1 - 1/p_j). This expression represents the probability that an integer is not divisible by any prime smaller than p_i, multiplied by its probability of being divisible by p_i.

The paper provides numerical evidence for n up to 1,000,000, showing that the actual prime counts stay strictly within the bounds defined by the recursive δ factors. These δ factors account for the accumulation of fractional parts in the floor functions. In analytic terms, these fractional parts are the discrete source of the oscillations that the Riemann Hypothesis predicts will cancel out over time. If the sum of these remainders grows significantly slower than the main term, it implies that the zeros of the zeta function must lie on the critical line to prevent anomalous growth in the prime counting function.

Error Terms and the Critical Line

The most profound connection to the Riemann Hypothesis lies in the treatment of the remainder term E(d;n) = [(n+1)²/d] - [n²/d] - (2n+1)/d. The source paper's recursive definition of the remainder, ε_bc, suggests that the "randomness" of primes is actually a highly constrained symmetry. In the context of the critical line, the oscillatory behavior of the error terms in the sieve corresponds to the imaginary parts of the zeta zeros.

If the Riemann Hypothesis were false, certain terms in the inclusion-exclusion expansion would fail to cancel, leading to a bias in the distribution of least prime factors. The source paper's ability to maintain tight bounds on π₂ across multiple scales suggests that the underlying recursive symmetry is preserved. This suggests that the distribution of primes in short intervals is as uniform as the Riemann Hypothesis requires.

Novel Research Pathways

1. Mobius Discrepancy and Square-Root Cancellation

A promising pathway involves rewriting the sieve function as a Mobius-weighted sum: S(n) = sum μ(d) E(d;n). By applying the fractional part analysis from arXiv:1406.0397v2, researchers can attempt to prove that this sum satisfies a square-root growth bound. This would provide a combinatorial proof of the non-vanishing of ζ(s) in the critical strip.

2. Spectral Analysis of Floor-Function Oscillations

The recursive identities can be viewed as the trace of a discrete operator. One could investigate whether the eigenvalues of an operator constructed from the C_j,i-1,k prime products cluster near the imaginary parts of the known zeta zeros. This would bridge the gap between the discrete sieve and the spectral theories of the Riemann Hypothesis.

Computational Implementation

(* Section: Sieve Density and Zeta-Zero Comparison *)
(* Purpose: Compare the recursive sigma function from arXiv:1406.0397v2 to Zeta zeros *)

Module[{nValues, primes, sigmaRec, piActual, zeros, plot1, plot2},
  nValues = Range[10, 500, 10];
  
  (* Recursive Sieve Density Function sigma(eta_n) *)
  sigmaRec[n_] := N[(2*n + 1) * Sum[
    (1/Prime[i]) * Product[1 - 1/Prime[j], {j, 1, i - 1}], 
    {i, 1, PrimePi[n + 1]}]];

  (* Actual Prime Count in Interval [n^2, (n+1)^2] *)
  piActual[n_] := PrimePi[(n + 1)^2] - PrimePi[n^2];

  (* Fetch first 15 imaginary parts of Zeta zeros *)
  zeros = Table[Im[ZetaZero[k]], {k, 1, 15}];

  (* Visualization of Sieve Error vs n *)
  plot1 = ListLinePlot[Table[{n, piActual[n] - sigmaRec[n]}, {n, nValues}], 
    PlotLabel -> "Sieve Discrepancy", 
    AxesLabel -> {"n", "Error"}, 
    PlotStyle -> Blue];

  (* Visualization of Zeta Zeros on the Critical Line *)
  plot2 = Plot[Abs[Zeta[1/2 + I*t]], {t, 0, 50}, 
    PlotLabel -> "Zeta on Critical Line", 
    AxesLabel -> {"t", "|Zeta(1/2+it)|"}, 
    ColorFunction -> "DarkRainbow"];

  Print[Column[{plot1, plot2}]]
]

Conclusions

The recursive framework in arXiv:1406.0397v2 offers a deterministic handle on the distribution of primes by focusing on the exact combinatorial structure of short intervals. By isolating the role of floor-function remainders, the paper identifies the discrete mechanism that governs prime density fluctuations. The most promising avenue for further research is the formal linkage of these recursive remainder terms to the square-root cancellation required by the Riemann Hypothesis. Ultimately, this work suggests that the distribution of primes is not a matter of chance, but a consequence of a rigid recursive symmetry that manifests most clearly on the critical line.

References

• arXiv:1406.0397v2: Recursive Formulas for Prime and Twin Prime Counts.
• arXiv:1305.0035: On the Distribution of Primes and the Riemann Hypothesis.
• Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function. Oxford University Press.
• Schoenfeld, L. (1976). Sharper bounds for the Chebyshev functions theta(x) and psi(x). II.