Primorial Bounds and the Nicolas Inequality: A Discrete Path to the Critical Line

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Introduction

The Riemann Hypothesis remains the most significant unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. While the hypothesis is fundamentally a statement about the distribution of complex zeros, its implications permeate the entirety of number theory, particularly the distribution of prime numbers. The source paper arXiv:hal-04168764v4, titled "The Riemann Hypothesis and the Nicolas Inequality," provides a rigorous exploration of the arithmetic equivalents of this conjecture, specifically focusing on the behavior of the sum-of-divisors function σ(n) and the primorial numbers N_n.

The motivation for this analysis stems from the historical work of Guy Robin and Jean-Louis Nicolas. Robin demonstrated that the Riemann Hypothesis is equivalent to the assertion that the sum-of-divisors function satisfies a specific inequality for all n > 5040. Similarly, Nicolas proved that the hypothesis is true if and only if a specific ratio involving the Euler totient function holds for all primorials. The specific problem addressed in arXiv:hal-04168764v4 involves the refinement of these bounds and the investigation of the Nicolas inequality as a primary diagnostic tool for the validity of the hypothesis.

Mathematical Background

To establish the foundation for our analysis, we define the primary arithmetic functions involved. The sum-of-divisors function, denoted as σ(n), is the sum of all positive divisors of n. The Riemann zeta function, ζ(s), is defined for Re(s) > 1 by the Dirichlet series and extends to a meromorphic function with a simple pole at s = 1. The connection between σ(n) and ζ(s) is established through the behavior of the ratio σ(n)/n.

As n increases, this ratio is bounded by the product of p/(p-1) for all primes p dividing n. Mertens' Second Theorem provides the asymptotic behavior of this product: the product of (1 - 1/p) inverse for p less than or equal to x is approximately e^γ log x, where γ is the Euler-Mascheroni constant. A critical structure used in arXiv:hal-04168764v4 is the primorial number N_n, defined as the product of the first n primes. Primorials are the most efficient "constructors" of large σ(n)/n ratios because they include the smallest possible primes, thereby maximizing the growth of the function.

Main Technical Analysis

Arithmetic Density and the Nicolas Inequality

The core of the analysis in arXiv:hal-04168764v4 revolves around the behavior of the function g(n), defined as the ratio of the prime product to the logarithmic bound e^γ log θ(p_n). The Riemann Hypothesis is equivalent to the statement that g(n) < 1 for all n > 1. The paper investigates the logarithmic derivative of this relation to determine the local extrema of the ratio.

Consider the transition from N_n to N_{n+1}. For the Nicolas inequality to hold, the increase in the numerator must be strictly compensated by the increase in the logarithmic bound. The source paper demonstrates that this compensation is directly tied to the error term in the Prime Number Theorem. Specifically, if θ(x) > x for any sufficiently large x, the Nicolas inequality is threatened. However, the Riemann Hypothesis implies that θ(x) stays very close to x, with a strictly controlled error term.

Spectral Properties and the Chebyshev Function

The analysis further connects the discrete jumps in the Chebyshev function θ(x) to the zeros of the zeta function. Using the explicit formula, θ(x) can be expressed as x minus the sum of (x^ρ / ρ) over the non-trivial zeros ρ. The source paper arXiv:hal-04168764v4 argues that the oscillations in θ(x) caused by these zeros determine whether the Nicolas ratio can ever exceed 1.

If any zero ρ has a real part greater than 1/2, the term x^ρ would grow faster than the square root of x, eventually causing θ(x) to exceed x by a margin large enough to violate the Nicolas inequality. The paper provides a rigorous bound for the product using the properties of the primorials, showing that for the inequality to fail, there must exist a peak in the distribution of primes where the local density significantly exceeds the average density predicted by the Prime Number Theorem.

Novel Research Pathways

Integration of Keiper-Li Coefficients: A promising direction involves connecting the Nicolas inequality with Li's criterion, which relates the positivity of certain coefficients λ_n to the Riemann Hypothesis. Developing a generating function that maps the jumps in the Nicolas ratio to these coefficients could provide a unified proof strategy.
Generalized Dedekind Functions: Research could investigate whether an analogue of the Nicolas inequality holds for the Dedekind psi function across different number fields. This would extend the findings of arXiv:hal-04168764v4 to the Generalized Riemann Hypothesis (GRH).
Computational Sieve for Abundant Numbers: Utilizing high-performance computing to check the Nicolas inequality for "Colossally Abundant" numbers—those that maximize the ratio σ(n)/n—could provide empirical verification for values of n exceeding 10^100.

Computational Implementation

(* Section: Nicolas Inequality Diagnostic *)
(* Purpose: To visualize the ratio g(n) = f(n) / (Exp[EulerGamma] * Log[Theta[p_n]]) *)

NicolasDiagnostic[maxN_] := Module[
  {primes, thetaValues, fValues, nicolasRatios},
  
  (* Generate the first maxN primes *)
  primes = Prime[Range[maxN]];
  
  (* Calculate the first Chebyshev function Theta(p_n) *)
  thetaValues = Accumulate[Log[primes]];
  
  (* Calculate the product f(n) = Product of p/(p-1) *)
  fValues = Accumulate[Log[primes / (primes - 1)]];
  fValues = Exp[fValues];
  
  (* Calculate the ratio g(n) relative to the Nicolas bound *)
  nicolasRatios = Table[
    fValues[[i]] / (Exp[0.5772156649] * Log[thetaValues[[i]]]),
    {i, 2, maxN}
  ];
  
  (* Visualization of the ratios *)
  ListLinePlot[nicolasRatios, 
    PlotRange -> {0.8, 1.1}, 
    AxesLabel -> {"n (Prime Index)", "g(n)"},
    PlotLabel -> "Nicolas Inequality Ratio g(n)",
    GridLines -> {None, {1.0}},
    PlotStyle -> Blue
  ]
];

(* Execute the diagnostic for the first 500 primorials *)
NicolasDiagnostic[500]

Conclusions

The analysis of arXiv:hal-04168764v4 reinforces the deep connection between the discrete world of prime products and the continuous world of the Riemann zeta function. By focusing on the Nicolas inequality, the paper provides a concrete arithmetic benchmark that is equivalent to the Riemann Hypothesis. The stability of the ratio g(n) is intrinsically linked to the non-violation of the Prime Number Theorem's optimal error term. Moving forward, the most promising avenue appears to be the extension of these primorial bounds to generalized L-functions, which would further constrain the possible locations of non-trivial zeros across a wider class of mathematical objects.

References

arXiv:hal-04168764v4
Nicolas, J. L. (1983). "Petites valeurs de la fonction d'Euler." Journal of Number Theory.
Robin, G. (1984). "Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann." Journal de Mathematiques Pures et Appliquees.