Primorial Oscillations and the Robin Criterion: A Technical Analysis of the Riemann Hypothesis Bridge

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Introduction

The Riemann Hypothesis (RH) remains the most significant unsolved problem in analytic number theory, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line where the real part is 1/2. While traditionally studied through the lens of complex analysis, the work of Jean-Louis Nicolas in arXiv:hal-00666154 highlights a powerful equivalent formulation in elementary number theory. This formulation centers on Robin's inequality, which states that the sum-of-divisors function σ(n) is bounded by e^γ n log log n for all n > 5040.

The core contribution of arXiv:hal-00666154 is the development of explicit, quantitative bounds for a specific function f(x) constructed from primorials (the product of the first k primes). By translating analytic information about the zeros of ζ(s) into usable error terms, this research provides a bridge between the discrete nature of prime products and the continuous properties of the zeta function. This article analyzes the refined bounds for the coefficient c(N_k), the impact of the zero-sensitive term W(x), and the implications for the distribution of prime powers.

Mathematical Background

The foundation of this analysis rests on the behavior of primorials N_k = p₁ * p₂ * ... * p_k. These numbers are central to testing the upper bounds of multiplicative functions because they maximize the ratio of divisors to the magnitude of the number itself. The primary object of study is the Nicolas function f(x), defined as f(x) = e^γ log log(N_k) φ(N_k)/N_k for p_k ≤ x < p_k+1, where φ is Euler's totient function.

Mertens' Second Theorem establishes the asymptotic behavior of the product of (1 - 1/p), which is inversely proportional to e^γ log x. The discrepancies from this asymptotic limit are governed by the distribution of the non-trivial zeros of ζ(s). The source paper utilizes Chebyshev functions θ(x) and ψ(x) to quantify these discrepancies. Specifically, θ(x) sums log p for primes p ≤ x, while ψ(x) counts all prime powers p^m ≤ x. The difference ψ(x) - θ(x) captures the contribution of squares, cubes, and higher powers of primes, which plays a critical role in the refined estimates of the Nicolas function.

Oscillatory Fluctuations and the Nicolas Coefficient

The technical core of arXiv:hal-00666154 involves the analysis of the coefficient c(N_k), which measures the deviation of the primorial Euler product from its asymptotic mean. The paper establishes a relationship where log f(p_k) is approximately (e^-γ / log log N_k) * (c(N_k) / sqrt(log N_k)). This identity demonstrates that the sign of log f(p_k)—and thus the validity of the equivalent Robin inequality—is controlled by the sign and magnitude of c(N_k).

One of the striking results is the determination of the extreme values of this coefficient. The paper proves that the limit inferior of c(n) is negative infinity and the limit superior is positive infinity. Despite these wild oscillations, the paper provides explicit bounds for c(N_k) in terms of a parameter β (representing the supremum of the real parts of the zeros minus 1/2). Under the Riemann Hypothesis, where β = 0, the bounds on c(N_k) are tight enough to ensure that the inequality f(n) < 1 is maintained for all sufficiently large n.

Prime Power Contributions and Chebyshev Bounds

To maintain the rigor of the inequality, the paper meticulously accounts for the difference between ψ(x) and θ(x). A significant lower bound is derived: ψ(x) - θ(x) ≥ θ(x^1/2) + θ(x^1/3). By further refining this as x^1/2 + x^1/3 - T(x^1/2) - T(x^1/3), where T is an explicit error function, the analysis ensures that the second-order terms provide a sufficient "buffer" against fluctuations.

Numerical verification in the source shows that the sum of these error terms, when normalized by x^1/3, is bounded by approximately 0.86157. This constant is crucial for proving that higher-order prime powers do not push the ratio f(n) over the threshold defined by the Euler-Mascheroni constant. This separation of smooth growth from oscillatory error is a hallmark of the technical depth found in arXiv:hal-00666154.

Zero Distribution and the W(x) Parameter

The term W(x) serves as a surrogate for the truncated sum over the non-trivial zeros of the zeta function. In the explicit formulae developed by Nicolas, W(x) packages the entire dependency on the Riemann Hypothesis into a single, controllable variable. If RH is true, W(x) is bounded by the small constant β. If RH were false, the existence of a zero with a real part greater than 1/2 would cause W(x) to grow as a power of x, eventually violating the established bounds.

The paper defines a specific threshold v₀ = 0.00000312 as the maximum allowable fluctuation consistent with current numerical evidence for RH. By bounding the auxiliary integrals J₁ and J₂, the analysis proves that all non-zero contributions (such as gamma-factor effects and smooth remainders) are strictly smaller than the term driven by W(x) once x is moderately large. This isolates the "RH-sensitive" component, allowing for a focused investigation into zero-free regions.

Novel Research Pathways

The structures presented in arXiv:hal-00666154 suggest several promising directions for future research:

• Distributional Modeling of c(N_k): Treating the zero sum underlying W(x) as a random trigonometric series could allow researchers to predict the frequency and magnitude of excursions of the Nicolas function. Using random matrix theory (GUE heuristics), one could model the variance of W(N_k) to predict the "near-misses" of Robin's inequality.
• Generalized Robin Criteria for Beurling Primes: Applying this explicit analysis to Beurling generalized prime systems could reveal whether the constants (0.07 and 3.7) found in the c(N_k) bounds are universal or specific to the standard integers.
• Quantitative First Counterexample Bounds: By combining the W(x) estimates with high-precision zero computations up to a specific height H, researchers can enlarge the range of unconditional verification for the Nicolas criterion, potentially moving the threshold for potential counterexamples to astronomically large values.

Computational Implementation

The following Wolfram Language code computes the Nicolas-style function f(N_k) on primorials and calculates the error term c(N_k) to visualize its oscillatory nature.

(* Section: Primorial Nicolas Analysis *)
(* Purpose: Compute f(Nk) and the error coefficient c(Nk) *)

Module[{kmax = 100, primes, logNk, Nk, phiOverNk, fNk, gammaVal, cNk, dataC},
  
  (* Generate primorial ingredients *)
  primes = Prime[Range[kmax]];
  logNk = Accumulate[Log[primes]];
  Nk = Exp[logNk];
  
  (* phi(Nk)/Nk = Product[(1 - 1/p)] over first k primes *)
  phiOverNk = Accumulate[Log[1 - 1/primes]] // Exp;
  
  (* Nicolas-style function: exp(gamma) * log log(Nk) * phi(Nk)/Nk *)
  gammaVal = N[EulerGamma, 20];
  fNk = Exp[gammaVal] * Log[logNk] * phiOverNk;
  
  (* Calculate c(Nk) based on the asymptotic log f(Nk) ~ (e^-gamma/loglog Nk)*(cNk/sqrt(log Nk)) *)
  (* Rearranging: cNk = log(fNk) * exp(gamma) * loglog(Nk) * sqrt(log Nk) *)
  cNk = Table[
    Log[fNk[[i]]] * Exp[gammaVal] * Log[logNk[[i]]] * Sqrt[logNk[[i]]],
    {i, 2, kmax}
  ];
  
  dataC = Transpose[{Range[2, kmax], cNk}];
  
  (* Plot the oscillation of the c(Nk) coefficient *)
  Print[ListLinePlot[dataC, 
    PlotLabel -> "Oscillation of Nicolas Coefficient c(Nk)", 
    AxesLabel -> {"k", "c(Nk)"},
    PlotStyle -> Red, 
    ImageSize -> Large]];
    
  Print["Maximum c(Nk) in range: ", Max[cNk]];
  Print["Minimum c(Nk) in range: ", Min[cNk]];
]

This implementation allows for the observation of how f(N_k) approaches the critical threshold and how the coefficient c(N_k) fluctuates, reflecting the underlying distribution of prime numbers and their relation to the zeta function's zeros.

Conclusions

The analysis of arXiv:hal-00666154 demonstrates that the Riemann Hypothesis is inextricably linked to the explicit bounds of multiplicative functions on primorials. By refining the error terms in Mertens' products and providing a rigorous treatment of the difference between prime and prime-power counting functions, Nicolas has established a quantitative framework for the Robin criterion. The most promising avenue for future work lies in the integration of verified zero-height data with the W(x) parameter to tighten the bounds on c(N_k), potentially leading to a definitive proof of the inequality across all integer ranges.

References

• Nicolas, J.-L. (2012). "On Robin's inequality". arXiv:hal-00666154
• Robin, G. (1984). "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann". Journal de Mathématiques Pures et Appliquées, 63, 187-213.
• Lagarias, J. C. (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". The American Mathematical Monthly, 109(6), 534-543.