Cubic Intervals and the Distribution of Primes: Bridging Local Gaps to the Riemann Hypothesis

Introduction

The distribution of prime numbers is a central theme in analytic number theory, acting as a bridge between additive and multiplicative structures. A fundamental challenge in this field is determining the maximum possible gap between consecutive primes, denoted as g_n = p_n+1 - p_n. While the Prime Number Theorem provides a global average for these gaps, it fails to provide deterministic upper bounds for specific intervals. The Riemann Hypothesis (RH) provides the most famous prediction in this regard, suggesting that the error term in the prime counting function is essentially square-root in nature, which would imply that g_n = O(p_n^1/2 log p_n).

The research paper arXiv:hal-02680213v1, authored by Madieyna Diouf, introduces a novel framework for analyzing these gaps by examining primes in relation to cubic intervals. By investigating the existence of primes between N³ and (N+1)³, the paper derives an unconditional bound of g_n = O(p_n^2/3). This result is significant because it provides a bridge between elementary counting methods and the deep analytic properties of the Riemann zeta function zeta(s).

This article provides a comprehensive technical analysis of Diouf's methodology, exploring the auxiliary parameters used to tether primes to square-proximate multiples. We further investigate how these local algebraic constraints can be integrated into the broader study of the zeta function's zeros on the critical line, proposing new research pathways that leverage these findings to advance our understanding of the Riemann Hypothesis.

Mathematical Background

To analyze the distribution of primes in short intervals, we must first establish the behavior of the prime gap g_n. Historically, the study of these gaps has progressed from the work of Hoheisel, who first proved a bound of the form p_n^{1 - epsilon}, to the refined exponents of Ingham, Montgomery, and Huxley. The bound O(p_n^2/3) is particularly interesting because it is the threshold required to prove the existence of primes between consecutive cubes.

The framework in arXiv:hal-02680213v1 utilizes a specific integer m and its square m² as a reference point. For a given prime p_n, we define αp_n as the largest odd multiple of p_n that does not exceed m². Similarly, for the next prime p_n+1, we define βp_n+1 as the largest odd multiple of p_n+1 that does not exceed m². The core of the paper's innovation lies in the introduction of a parameter c_n, which relates these multiples through the identity:

• αp_n = p_n(p_n+1 + 2c_n)
• βp_n+1 = p_n+1(p_n + 2c_n+1)

These parameters measure the "slack" or distance between the prime product and the square boundary m². By establishing that c_n is a non-increasing sequence (c_n+1 ≤ c_n), the paper provides a mechanism to bound the growth of the prime gap relative to the magnitude of the primes themselves.

Main Technical Analysis

Spectral Properties and Zero Distribution

The relationship between prime gaps and the Riemann zeta function is most clearly seen through the Explicit Formula. This formula relates the Chebyshev function ψ(x), which counts primes weighted by their logarithms, to the non-trivial zeros ρ of zeta(s). Specifically, ψ(x) is approximately x - sum(x^ρ/ρ). If the gaps g_n are large, it implies significant oscillations in ψ(x), which in turn necessitates that the zeros ρ are distributed away from the critical line Re(s) = 1/2.

The bound g_n = O(p_n^2/3) effectively restricts the possible frequencies and amplitudes of these oscillations. In the context of arXiv:hal-02680213v1, the parameter c_n acts as a local proxy for the density of primes. If c_n remains small, the primes are regularly spaced, supporting the hypothesis that the zeros of the zeta function are well-behaved. The analysis shows that the stability of c_n is linked to the distribution of quadratic residues, suggesting a deep connection between spectral properties and modular arithmetic.

The Bootstrap Mechanism for Cubic Gaps

The derivation of the 2/3 exponent relies on a bootstrapping technique. First, the paper establishes the inequality g_n < 2 sqrt(2p_n(c_n + 1)). This is derived by considering the midpoint m = (p_n + p_n+1)/2 and noting that m² - p_np_n+1 = (g_n/2)². Since αp_n is the largest odd multiple below m², the next multiple must exceed m², leading to a constraint on g_n in terms of c_n.

Second, the paper proves that c_n < p_n/g_n. This bound is obtained by analyzing the difference βp_n+1 - αp_n. By combining these two inequalities, we obtain:

• g_n² < 8p_n(c_n + 1)
• g_n² < 8p_n(p_n/g_n + 1)
• g_n³ < 8p_n² + 8p_ng_n

For large p_n, this simplifies to g_n = O(p_n^2/3). This result is robust and provides an unconditional guarantee of primes in cubic intervals, as the distance between N³ and (N+1)³ is roughly 3N², which matches the p_n^2/3 growth rate.

Novel Research Pathways

1. The Explicit Formula and Local Oscillations

A promising research direction involves integrating the c_n parameter directly into the sum over zeros in the explicit formula. By modeling c_n as a function of the imaginary parts of the zeros γ, one could potentially show that the non-increasing nature of c_n implies a lower bound on the density of zeros on the critical line. This would transform a local algebraic result into a global analytic constraint on zeta(s).

2. Arithmetic Progressions and Modular Dynamics

The definition of αp_n as a largest odd multiple suggests that the distribution of primes is constrained by the dynamics of arithmetic progressions modulo p_n. Investigating the correlation between the sequence c_n and the distribution of quadratic residues could reveal new symmetries in the prime sequence. This pathway could utilize Sieve Theory to refine the bounds on c_n, potentially pushing the exponent below 2/3 toward the 1/2 predicted by RH.

3. Generalization to Higher-Order Intervals

The methodology used for cubic intervals (N³) can be generalized to arbitrary powers N^k. By defining c_n relative to m^k-1, researchers can investigate if the monotonicity of the correction parameter holds in higher dimensions. This would provide a systematic way to attack the generalized Legendre conjecture, proving that there is always a prime between n^k and (n+1)^k for all k, which is a significant consequence of the Riemann Hypothesis.

Computational Implementation

(* Section: Empirical Analysis of the Diouf Bound *)
(* Purpose: Calculate the auxiliary parameter c_n and verify the O(p_n^2/3) gap bound *)

AnalyzePrimeGaps[nMax_Integer] := Module[
  {primes, results, p1, p2, gn, m, mSq, alpha, cn, bound, p23},
  
  (* Generate primes and calculate parameters *)
  primes = Prime[Range[1, nMax + 1]];
  results = Table[
    p1 = primes[[i]];
    p2 = primes[[i + 1]];
    gn = p2 - p1;
    m = (p1 + p2)/2;
    mSq = m^2;
    
    (* Find largest odd multiple of p1 <= m^2 *)
    alpha = p1 * Floor[mSq/p1];
    If[EvenQ[alpha/p1], alpha = alpha - p1];
    
    (* Solve alpha = p1(p2 + 2cn) for cn *)
    cn = (alpha/p1 - p2)/2;
    
    (* Calculate the bound g_n < 2 * Sqrt[2 * p1 * (cn + 1)] *)
    bound = 2 * Sqrt[2 * p1 * (cn + 1)];
    p23 = p1^(2/3);
    
    {p1, gn, bound, p23},
    {i, 1, nMax}
  ];
  
  (* Plot results: Red is actual gap, Blue is 2/3 bound *)
  ListLinePlot[
    {results[[All, {1, 2}]], results[[All, {1, 4}]]},
    PlotLegends -> {"Actual Gap g_n", "Bound p_n^(2/3)"},
    AxesLabel -> {"Prime p_n", "Gap Size"},
    PlotLabel -> "Verification of Prime Gap Distribution in Cubic Intervals",
    PlotStyle -> {Red, Blue}
  ]
]

(* Execute for the first 1000 primes *)
AnalyzePrimeGaps[1000]

Conclusions

The analysis of prime gaps in cubic intervals provides a powerful deterministic tool for understanding the distribution of primes. By utilizing the c_n parameterization established in arXiv:hal-02680213v1, we can rigorously prove the existence of primes in intervals where classical methods often struggle. The derivation of the O(p_n^2/3) bound is not only an algebraic success but also a significant step toward reconciling local prime behavior with the global predictions of the Riemann Hypothesis.

The most promising avenue for future research lies in refining the statistical properties of c_n. If it can be shown that c_n grows at a logarithmic rate rather than a power rate, the gap bound would improve significantly, potentially leading to a proof of the Riemann Hypothesis's implications for prime distribution. Continued integration of computational verification and analytic theory remains essential for navigating the complexities of the critical line.

References

• Diouf, M. (2020). Primes between two consecutive cubes. arXiv:hal-02680213v1
• Baker, R. C., Harman, G., & Pintz, J. (2001). The difference between consecutive primes, II. Proceedings of the London Mathematical Society.
• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function. Proceedings of Symposia in Pure Mathematics.
• Vega, F. (2024). Robin's criterion on divisibility and the Riemann Hypothesis. arXiv:hal-04580042