Primorial Bounds and the Sum of Divisors: Proving the Riemann Hypothesis via Robin Inequality

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Introduction

The Riemann Hypothesis (RH) 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 problem is traditionally framed within the context of complex analysis and the spectral theory of operators, a parallel lineage of research has sought to establish the hypothesis through elementary inequalities involving arithmetic functions. The source paper arXiv:hal-04441541v4, authored by Frank Vega, contributes to this latter tradition by focusing on the behavior of the sum-of-divisors function σ(n) and its relationship to Robin's inequality.

The motivation for this analysis stems from the 1984 work of Guy Robin, who proved that the Riemann Hypothesis is equivalent to the assertion that σ(n) < e^γ * n * log log n for all n > 5040, where γ is the Euler-Mascheroni constant. This equivalence transformed a problem of complex analytic zeros into a problem of bounding the growth of a fundamental number-theoretic function. The paper arXiv:hal-04441541v4 investigates specific constraints on potential counterexamples to Robin's inequality, particularly focusing on the role of primorial numbers and the Chebyshev function θ(x).

Mathematical Background

To understand the arguments presented in arXiv:hal-04441541v4, we must define the primary mathematical objects and the established theorems that form the foundation of this research.

The Sum-of-Divisors Function: The function σ(n) is defined as the sum of all positive divisors of n. For a prime power p^a, σ(p^a) = (p^a+1 - 1) / (p - 1). Due to its multiplicativity, for n = ∏ p_i^a_i, σ(n) = ∏ (p_i^a_i+1 - 1) / (p_i - 1).
Robin's Theorem: Robin proved that if the Riemann Hypothesis is true, then for all n > 5040, the inequality σ(n) / n < e^γ * log log n holds. Conversely, if the Riemann Hypothesis is false, there exist infinitely many n for which this inequality is violated.
Nicolas's Inequality: Jean-Louis Nicolas proved a similar criterion involving primorials. Let N_k = p₁ * p₂ * ... * p_k be the k-th primorial. RH is true if and only if for all k ≥ 1, σ(N_k) / N_k < e^γ * log log N_k.
The Chebyshev Function: The first Chebyshev function θ(x) is defined as the sum of log p for all primes p ≤ x. Its behavior is intimately tied to the distribution of primes; the Prime Number Theorem is equivalent to the statement θ(x) ~ x.

Main Technical Analysis

The Primorial Ratio and Robin's Criterion

The core of the analysis in arXiv:hal-04441541v4 involves the evaluation of the function f(n) = σ(n) / (n * log log n). For the Riemann Hypothesis to hold, f(n) must remain below e^γ for all n > 5040. The paper focuses on a specific class of integers known as colossally abundant numbers. These numbers are the natural candidates for counterexamples because they maximize the ratio σ(n)/n relative to the size of n.

The author examines the behavior of f(n) as n approaches infinity by considering the prime factorization of n. If n is a primorial N_k, the ratio can be expressed as a product over primes. Using the property that (1 + 1/p) < (1 - 1/p)^-1, we can bound this by the Mertens product. The paper refines this by analyzing the error term in the approximation of log log N_k. Since log N_k = θ(p_k), the term log log N_k is approximately log θ(p_k). The analysis demonstrates that the convergence of θ(p_k) to p_k is fast enough to maintain the inequality, provided there are no unexpected gaps in the distribution of primes.

Asymptotic Bounds and the Critical Line

A significant portion of the technical derivation involves the connection between the growth of σ(n) and the behavior of the Mertens function M(x). The derivation shows that if σ(n)/n were to exceed the Robin bound, it would imply an oscillation in the distribution of primes π(x) of a magnitude √x * log x, which would necessitate the existence of zeros of ζ(s) with real part greater than 1/2.

The paper provides a rigorous proof-sketch showing that for any n that is a colossally abundant number, the ratio R(n) is strictly decreasing for sufficiently large n. The mathematical argument relies on replacing the sum over reciprocals of primes with log log p_k + B + o(1) using Mertens' theorems and showing that the difference is dominated by the term (p_k - θ(p_k)) / (p_k log p_k).

Novel Research Pathways

1. Spectral Interpretation of Robin's Ratio

A promising direction involves mapping the Robin ratio R(n) onto the eigenvalues of a specific operator. If one can construct a Hilbert space where the failure of Robin's inequality corresponds to the existence of an eigenvalue outside a specific unit disk, the problem could be solved using functional analysis.

2. Beurling Generalized Primes and Robin's Inequality

The source paper arXiv:hal-04441541v4 relies on the standard sequence of primes. However, the Riemann Hypothesis can be generalized to Beurling systems of generalized primes. Investigating whether Robin's inequality holds for a generalized number system if and only if the corresponding Beurling zeta function satisfies RH would identify the minimum conditions on prime density π(x) required for Robin's criterion to remain valid.

Computational Implementation

The following Wolfram Language code allows for the verification of Robin's inequality and the visualization of the ratio R(n) for primorial numbers, which are the most likely candidates for violating the hypothesis.

(* Section: Robin's Inequality and Primorial Analysis *)
(* Purpose: To visualize the ratio sigma(n)/(n log log n) against the e^gamma limit *)

(* Define the Robin Ratio function *)
robinRatio[n_] := DivisorSigma[1, n] / (n * Log[Log[n]]);

(* Generate the first k primorials *)
generatePrimorials[k_] := Table[Product[Prime[i], {i, 1, j}], {j, 3, k}];

(* Calculate ratios for the first 100 primorials *)
kMax = 100;
primorials = generatePrimorials[kMax];
ratios = Table[{Log[primorials[[i]]], robinRatio[primorials[[i]]]}, {i, 1, Length[primorials]}];

(* Define the e^gamma constant *)
expGamma = Exp[EulerGamma];

(* Visualization *)
PlotRobinRatio := ListPlot[ratios, 
    PlotStyle -> Blue, 
    Joined -> True, 
    PlotRange -> All, 
    AxesLabel -> {"log(n)", "R(n)"}, 
    PlotLabel -> "Robin Ratio for Primorials vs e^gamma",
    Epilog -> {Red, Dashed, InfiniteLine[{0, expGamma}, {1, 0}]}
];

Print[PlotRobinRatio];

(* Check for any violations in the first 500 primorials *)
checkViolations[k_] := Module[{v = {}},
    Do[
        n = Product[Prime[i], {i, 1, j}];
        If[robinRatio[n] > expGamma, AppendTo[v, j]],
        {j, 3, k}
    ];
    v
];
Print["Violations found at k indices: ", checkViolations[500]];

Conclusions

The analysis of arXiv:hal-04441541v4 provides a compelling look at the Riemann Hypothesis through the lens of elementary arithmetic functions. By focusing on the growth of the sum-of-divisors function at primorial numbers, the paper reinforces the deep connection between the distribution of primes and the upper bounds of σ(n). The technical derivation suggests that the constraints imposed by the Prime Number Theorem on the Chebyshev function are sufficient to prevent any violation of Robin's inequality, which in turn implies the validity of the Riemann Hypothesis.

References

arXiv:hal-04441541v4
Robin, G. (1984). Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. Journal de Mathématiques Pures et Appliquées, 63, 187-213.
Nicolas, J. L. (1983). Petites valeurs de la fonction d'Euler. Journal of Number Theory, 17(3), 375-388.