Sieve Theoretic Constraints and Critical Line Behavior in Prime Distribution

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Executive Summary

The research presented in arXiv:1504.01645v1 develops a sophisticated sieve-and-congruence construction that provides a novel framework for understanding prime density. By engineering a large "good modulus" P(x) and imposing layered congruence conditions, the paper identifies mechanisms to suppress the occurrence of primes in target intervals. The technical core involves an upper-bound sieve estimate controlled by Euler-type products and remainder terms bounded uniformly on square-free moduli. The primary connection to the Riemann Hypothesis (RH) lies in how these main terms are governed by Mertens-type products and weighted prime sums. Under the assumption of RH, the error terms associated with these products would tighten significantly, allowing for a more deterministic verification of prime densities in sparse sequences. This article synthesizes these findings to propose new pathways for investigating the zeros of the zeta function through the lens of sieve-theoretic stability.

Introduction

The distribution of prime numbers remains a central challenge in analytic number theory. While the Prime Number Theorem provides a global density, the behavior of primes in restricted subsets—such as Romanov-type sequences involving the sum of a prime and a power of two—requires refined tools. The source paper, arXiv:1504.01645v1, addresses this by constructing intervals where primes are forced to be rare through structured congruence systems. The problem is fundamentally one of additive number theory governed by the multiplicative properties of the integers involved.

The contribution of this analysis is to bridge the gap between the sieve bounds in the source paper and the structural requirements of the Riemann Hypothesis. The error terms in the sieve, denoted as R_d, represent the fluctuations in prime distribution across arithmetic progressions. If the zeros of the zeta function are confined to the critical line Re(s) = 1/2, these fluctuations are minimized, tightening the bounds on T(v)—the quantity representing the prime-generating capacity of the sequence. This approach translates the analytic properties of the Riemann zeta function into concrete additive bounds for binary linear equations involving primes.

Mathematical Background

The foundation of the sieve framework in arXiv:1504.01645v1 rests on the sieve function S(A, P, z), which counts elements in a set A not divisible by any prime in P up to limit z. A crucial component is the product formula derived from Mertens' Third Theorem: product over p ≤ w of (1 - 1/p) = (e^-gamma / log w) (1 + O(1/log w)), where gamma is the Euler-Mascheroni constant. This product represents the baseline probability that a random integer avoids small prime factors.

The paper introduces a density function rho(p) to account for excluded residues modulo p. The upper bound for prime values is given by: T(v) ≤ P(x)^D-1 ∏ (1 - 1/p) ∏ (1 - rho(p)/p). The Riemann zeta function is linked to these products through the Euler Product formula. The Riemann Hypothesis asserts that the error term in the Prime Number Theorem is O(x^1/2 log x), which in sieve theory translates to the ability to push the "level of distribution" D to higher thresholds while maintaining control over the remainder terms.

Main Technical Analysis

Sieve Bounds and Prime Density Suppression

The technical core involves partitioning an interval into subintervals and applying layered congruence conditions. For primes in specific sets, multiple residues are excluded, such as n ≡ 0 (mod p) and n ≡ 1 - 2^k (mod p). This dual constraint creates a filtration that preserves multiplicative structure while imposing additive constraints. The convergence of the resulting product is sensitive to the distribution of 2^k modulo p. Under the Generalized Riemann Hypothesis (GRH), the predictability of these residues allows for a sharper estimation of the product over sparse prime sets.

The Remainder Term and the Critical Line

The effectiveness of the sieve depends on the remainder term R_d. The source paper assumes |R_d| ≤ omega(d) for square-free integers. If zeros of the zeta function were to exist off the critical line, primes would "clump" in certain arithmetic progressions, making the remainder terms large and the resulting sieve bounds loose. The Riemann Hypothesis guarantees that primes are distributed as randomly as possible modulo d, ensuring that R_d remains small on average. This uniformity prevents the "few prime" scenario described in the source paper from occurring too frequently, providing a robust lower bound for the existence of primes in the sequence.

Spectral Interpretations and Good Moduli

The concept of a "good modulus" P(x) in arXiv:1504.01645v1 refers to a modulus for which primes are well-distributed among residue classes. This is the territory of Bombieri-Vinogradov type theorems. Under RH, the error terms for these distributions are essentially square-root suppressed. The spectral gap between the eigenvalues of the matrix M (with P(x)^D-1 rows) used in the paper controls the rate of convergence in the prime number theorem for sieve-selected integers. A large spectral gap implies rapid convergence, which is necessary to establish zero-free regions for the associated L-functions.

Novel Research Pathways

1. Sieve-to-Zeros Stability Framework

Formulation: Construct a family of admissible prime subsets for which the associated Euler products have precise iterated-log asymptotics. Connection: If these products demonstrate stable asymptotics with power-saving error terms, it may force RH-type constraints on the Dirichlet series built from those primes. Methodology: Compare the logarithmic derivatives of these series to the weighted prime sums in the sieve hypothesis to translate remainder terms into zero sums.

2. GRH-Optimized Construction

Formulation: Re-optimize the parameters of the arXiv:1504.01645v1 construction (interval length y and dimension D) under the explicit assumption of GRH. Connection: GRH provides near square-root cancellation for primes in progressions, tightening the distributional inputs. Methodology: Rewrite each sieve remainder in terms of discrepancies in residue classes and track how the GRH error term propagates through the upper-bound sieve to find shorter windows of prime rarity.

3. Resonance Framework for Sieve Weights

Formulation: Design sieve weights that "resonate" with the oscillatory terms in the explicit formula for the prime counting function. Connection: If zeros off the critical line existed, they would create oscillations that a properly tuned sieve could amplify. Methodology: Express sieve remainders as sums over arithmetic progressions and maximize sensitivity to hypothetical zeros with Re(s) > 1/2 to create a diagnostic contradiction.

Computational Implementation

Wolfram Language

(* Section: Sieve Density and Zeta Zero Correlation *)
(* Purpose: Compare prime Euler products with Mertens asymptotics and display zeta zeros *)

Module[
  {xs, mertensProd, mertensMain, ratioAll, zeros, gamma, maxRange, romanovPrimes},

  gamma = EulerGamma;
  xs = {10^2, 10^3, 10^4, 10^5};
  maxRange = 1000;

  (* Calculate exact Mertens product and theoretical main term *)
  mertensProd[x_] := N[Times @@ (1 - 1/# & /@ Prime[Range[PrimePi[x]]]), 20];
  mertensMain[x_] := N[Exp[-gamma]/Log[x], 20];

  ratioAll = Table[
    {x, mertensProd[x], mertensMain[x], mertensProd[x]/mertensMain[x]},
    {x, xs}
  ];

  (* First 10 zeta zeros on the critical line *)
  zeros = Table[ZetaZero[n], {n, 1, 10}];

  (* Romanov-type sequence density check: p + 2^k *)
  romanovPrimes = Select[Flatten[Table[Prime[i] + 2^k, {i, 1, 100}, {k, 1, 8}]], 
     # <= maxRange && PrimeQ[#] &];

  Print["Table: {x, Product_{p<=x}(1-1/p), e^{-gamma}/log x, ratio}"];
  Print[Grid[Prepend[ratioAll, {"x", "prod", "main", "ratio"}], Frame -> All]];

  Print["\nFirst 10 nontrivial zeros of Zeta(s): ", zeros];
  
  Print["\nRomanov Prime Density Sample (p + 2^k): ", N[Length[romanovPrimes]/maxRange]];

  (* Plot of Mertens ratio to visualize convergence *)
  Plot[
    mertensProd[t]/mertensMain[t], {t, 100, 5000}, 
    PlotLabel -> "Ratio of Mertens Product to Asymptotic Main Term",
    AxesLabel -> {"x", "Ratio"}
  ]
]

arXiv:1504.01645v1 demonstrates that prime-suppressing constructions are driven by quantified upper-bound sieve inequalities. The connection to the Riemann Hypothesis is structural; any sharpening of product asymptotics in short ranges ultimately depends on the control of prime distribution errors governed by the zeros of the zeta function. The most promising next step is a GRH-optimized re-derivation of the paper's parameter tradeoffs to isolate how far the construction can be pushed when one has RH-level error terms for primes in arithmetic progressions.

References

arXiv:1504.01645v1
Montgomery, H. L., and Vaughan, R. C. (2006). Multiplicative Number Theory I: Classical Theory. Cambridge University Press.
Iwaniec, H., and Kowalski, E. (2004). Analytic Number Theory. American Mathematical Society.