Prime Existence in Linear Intervals: A Combinatorial Path to the Riemann Hypothesis

Executive Summary

The research presented in arXiv:1706.01009v1 establishes a rigorous combinatorial framework for proving the existence of prime numbers within specific linear intervals, such as [4n, 5n], by leveraging refined estimates of factorial ratios and binomial coefficients. By constructing explicit upper and lower bounds for these combinatorial structures using sharpened Stirling-type approximations, the paper demonstrates that the product of primes in a target interval must exceed unity for sufficiently large n. This approach provides a bottom-up methodology to constrain the fluctuations of the prime-counting function, which are fundamentally governed by the distribution of zeros of the Riemann zeta function. The connection to the Riemann Hypothesis (RH) is established through the requirement that the error terms in the Prime Number Theorem remain within the square-root growth bounds predicted by the hypothesis. This combinatorial "forcing" mechanism offers a promising avenue for translating analytic density requirements into computable, finite inequalities that can be verified for large parameter ranges.

Introduction

A central objective in number theory is to prove that every sufficiently large interval of a specific shape contains at least one prime. This lineage of research began with Bertrand’s Postulate, which stated that for every n > 1, there exists a prime p such that n < p < 2n. Chebyshev later proved this using properties of the central binomial coefficient C(2n, n). However, proving the existence of primes in narrower or shifted intervals, such as [4n, 5n], requires much more sophisticated control over the prime power decomposition of factorial ratios.

The paper arXiv:1706.01009v1 contributes to this field by providing explicit, non-asymptotic bounds for generalized binomial factors. These bounds are essential because the Prime Number Theorem (PNT) only describes the average density of primes and does not guarantee their existence in specific short ranges unless the error term is strictly bounded. Since the magnitude of this error term is the core of the Riemann Hypothesis, the combinatorial methods used to establish prime existence in short intervals provide a direct probe into the regularity of prime distribution.

Mathematical Background

The technical core of the source paper involves the analysis of the prime power valuation of binomial coefficients. According to Legendre's formula, the exponent of a prime p in n! is the sum of floor(n/p^i). The author defines a structure B based on rational ratios:

B = delta(4n/3, 5n/3) * C(floor(5n/3), floor(4n/3))

To bound such terms, the author introduces refined Stirling-type functions u(n) and l(n) such that l(n) ≤ n! ≤ u(n). These functions are defined as:

• l(n): sqrt(2 * pi * n) * (n/e)^n * exp(1 / (12n + 1))
• u(n): sqrt(2 * pi * n) * (n/e)^n * exp(1 / 12n)

These bounds allow for the derivation of explicit inequalities for prime products. For example, the paper demonstrates that B < 5.153158 * n^(1/2) * (3125/256)^(n/3). In analytic number theory, these combinatorial products are related to the Chebyshev functions theta(x) and psi(x). The Riemann Hypothesis is equivalent to the assertion that |theta(x) - x| is bounded by a term of order sqrt(x) log(x) squared. Consequently, proving that a prime product in a specific interval is greater than 1 is equivalent to proving that the difference theta(x_2) - theta(x_1) remains positive, which is a key requirement of RH.

Main Technical Analysis

Sieve Bounds and Prime Density

The argument relies on a "forcing architecture" where a large integer is constructed such that its prime factorization must include primes from a target interval. The paper defines the product T_2 as the product of primes p in the range sqrt(6n) < p ≤ 5n. This product is further partitioned into sub-intervals to isolate the contribution of primes in ranges like [n, 6n/5], [5n/4, 3n/2], and [4n, 5n].

The critical insight is that the exponential growth rate of the binomial coefficients, characterized by bases such as (3125/256) or (9^9 / 8^8), acts as an "energy source." If one can prove that primes outside the target interval cannot account for the total magnitude of the binomial coefficient, then at least one prime must exist within the target range. This is expressed through the inequality:

log(T_2) > (n/log n) * (0.0170459 - 13 log n / n - 3.76518 / sqrt(n))

As n increases, the positive constant term dominates the negative terms, ensuring that the prime product exceeds 1. This provides a discrete, combinatorial analog to the analytic density proofs that rely on the non-vanishing of the zeta function along the critical line.

Comparison with the Weak Prime Number Theorem

The source paper includes a comparison between its combinatorial results and the "Weak PNT," which provides explicit bounds on the prime-counting function without assuming RH. For n = 10^8, the paper's method yields a lower bound for the number of primes that is positive and growing, though still smaller than the actual count pi(5n) - pi(4n). This gap represents the "slack" that the Riemann Hypothesis would theoretically tighten. Under RH, the fluctuations in prime density are minimized, meaning that the combinatorial bounds would be much closer to the actual counts, allowing for even shorter intervals to be proven.

Novel Research Pathways

1. Optimization of Splitting Parameters via Entropy

The constants in the paper, such as 4n/3 and 5n/3, are chosen to facilitate the proof of prime existence. A potential research direction is to treat these as variables (alpha, beta) and use continuous entropy methods to find the optimal ratios that maximize the lower bound of the prime product. Under the assumption of the Riemann Hypothesis, these parameters could be pushed to their theoretical limits, potentially proving prime existence in intervals of length n^(1/2) using purely combinatorial templates.

2. Binomial-Spectral Identities

One can investigate the mapping between the p-adic valuation of the generalized binomial coefficients used in arXiv:1706.01009v1 and the zeros of the zeta function. Since the valuation detects prime presence, and the prime distribution is governed by the zeros, there should exist an identity connecting the sum of log(p) over the binomial factors to a sum over the zeros rho. This would create a new "combinatorial RH equivalence" phrased in terms of factorial ratios.

Computational Implementation

The following Wolfram Language code demonstrates the relationship between the combinatorial bounds described in the paper and the actual distribution of primes, while also visualizing the zeta zeros that govern the underlying density.

(* Section: Combinatorial Prime Density and Zeta Correlation *)
(* Purpose: To verify the existence of primes in [4n, 5n] and relate to RH *)

Module[{nValues, results, thetaGap, rhScale, zeros},
  (* Define the Chebyshev theta gap for the interval [4n, 5n] *)
  thetaGap[n_] := ChebyshevTheta[5*n] - ChebyshevTheta[4*n];
  
  (* Define the RH-motivated fluctuation scale sqrt(n) * Log[n]^2 *)
  rhScale[n_] := N[Sqrt[n] * Log[n]^2];
  
  (* Sample values of n across decades *)
  nValues = {10^4, 10^5, 10^6, 10^7};
  
  Print["n | Theta Gap | RH Fluctuation Scale | Ratio"];
  
  results = Table[
    {n, thetaGap[n], rhScale[n], N[thetaGap[n]/rhScale[n]]},
    {n, nValues}
  ];
  
  (* Output result table *)
  Print[Grid[Prepend[results, {"n", "theta(5n)-theta(4n)", "sqrt(n)log(n)^2", "Ratio"}], Frame -> All]];

  (* Visualize the gap growth compared to the fluctuation scale *)
  Print[LogLinearPlot[{
    ChebyshevTheta[5*x] - ChebyshevTheta[4*x],
    Sqrt[x] * Log[x]^2
  }, {x, 10^3, 10^6}, 
  PlotLegends -> {"Actual Prime Product (Log)", "RH Fluctuation Bound"},
  AxesLabel -> {"n", "Magnitude"}]];

  (* Sample the first few Zeta Zeros to emphasize the spectral driver *)
  zeros = Table[ZetaZero[k], {k, 1, 5}];
  Print["First 5 Zeta Zeros (Critical Line): ", zeros];
]

Conclusions

The analysis of arXiv:1706.01009v1 demonstrates that explicit bounds on factorial ratios provide a robust mechanism for guaranteeing prime existence in linear intervals. By translating these combinatorial requirements into the language of Chebyshev functions, we see a clear connection to the Riemann Hypothesis: the positivity of the prime products is a discrete manifestation of the density constraints imposed by the zeta function’s zeros. The most promising avenue for further research lies in the optimization of these binomial structures using RH-conditional error terms to shrink the proven intervals toward the square-root limit.

References

• arXiv:1706.01009v1: "On the existence of prime numbers in certain intervals."
• arXiv:1511.04571v2: "Refined Stirling's approximation and its applications in number theory."
• arXiv:1004.1563v2: "Sieve procedure for the Mobius prime-functions."