The Arithmetic Frontier: Proving the Riemann Hypothesis Through Divisor Sums and Primorials

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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, denoted as zeta(s), possess a real part equal to 1/2. While the hypothesis is typically framed within the context of complex analysis, a parallel track of investigation exists within elementary number theory. This track translates the analytic properties of zeta(s) into inequalities involving arithmetic functions, most notably the sum-of-divisors function, sigma(n), and the Euler totient function, phi(n).

The research presented in hal-04454257v1 focuses on these arithmetic equivalences, specifically the Robin criterion and the Nicolas criterion. These criteria provide a rigorous framework to test the boundaries of the critical strip through the behavior of primorials and highly composite numbers. The motivation for this analysis stems from the observation that the distribution of prime numbers is inextricably linked to the fluctuations of the abundance of integers.

Mathematical Background

To understand the contributions of hal-04454257v1, one must first define the primary mathematical objects that link arithmetic to the Riemann zeta function. The sum-of-divisors function, sigma(n), is defined as the sum of all positive divisors of n. The ratio f(n) = sigma(n) / n represents the abundance of an integer. In 1984, Guy Robin proved that the Riemann Hypothesis is equivalent to the assertion that sigma(n) is less than exp(gamma) * n * log(log(n)) for all n greater than 5040, where gamma is the Euler-Mascheroni constant.

A related structure is the primorial, N_k, which is the product of the first k primes. The Nicolas criterion states that the Riemann Hypothesis is true if and only if for all k greater than or equal to 1, the ratio N_k / phi(N_k) is greater than exp(gamma) * log(log(N_k)). These relationships are mediated by Mertens' Third Theorem, which connects the product of prime factors to the logarithm of the largest prime involved.

Asymptotic Expansions and Primorial Extremals

The core technical contribution of the source paper involves the analysis of the function G(n) = sigma(n) / (n * log(log(n))). To evaluate the hypothesis via this route, researchers identify Colossally Abundant (CA) numbers or Superabundant numbers that maximize this ratio. The analysis begins with the relationship between the sum-of-divisors function and the prime distribution, using the logarithmic derivative and Chebyshev functions.

The explicit formula connects the sum over primes to the sum over the zeros of the zeta function. If the Riemann Hypothesis is true, the error term in the distribution of primes is tightly bounded. The source paper hal-04454257v1 refines these error terms using the explicit formula for psi(x). By applying the Rosser-Schoenfeld bounds, the paper establishes that any violation of the Nicolas inequality must occur for relatively small k, or else require a massive failure of the prime distribution that would contradict known zero-free regions.

Furthermore, the paper utilizes sieve methods to bound the prime factors of a potential counterexample. It is demonstrated that if a counterexample exists, the number of distinct prime factors must be exceptionally large, pushing the search far beyond the reach of current brute-force computational methods.

Novel Research Pathways

The analysis opens several distinct avenues for further research, particularly in the integration of arithmetic inequalities with the spectral theory of L-functions.

Generalized Robin Criteria: Extending these criteria to Dirichlet L-functions by investigating twisted sum-of-divisors functions. This would provide a way to verify the Generalized Riemann Hypothesis for specific characters through finite arithmetic checks.
Information-Theoretic Bounds: Applying concepts of Kolmogorov complexity and Shannon entropy to the sequence of divisors of Colossally Abundant numbers. The rate of growth of divisor entropy is linked to the smoothness of the prime distribution.
Beurling Zeta Functions: Investigating generalized prime systems that behave like primes but may violate the Robin inequality. This allows for a quantitative stability theorem showing how much the prime distribution can fluctuate before the hypothesis is invalidated.

Computational Implementation

To visualize the relationship between the sum-of-divisors function and the Riemann Hypothesis, the following Wolfram Language script calculates the Robin ratio for primorial numbers and checks the proximity to the theoretical limit.

(* Section: Robin Criterion and Primorial Analysis *)
(* Purpose: Calculate G(n) = sigma(n)/(n log log n) for primorials *)

Module[{primes, primorials, robinRatios, eGamma, kMax, plot},
  kMax = 100;
  primes = Table[Prime[i], {i, kMax}];
  primorials = FoldList[Times, primes];
  
  robinRatio[n_] := DivisorSigma[1, n] / (n * Log[Log[n]]);
  eGamma = Exp[EulerGamma] // N;
  
  (* N_7 = 510510 is the first primorial > 5040 *)
  robinRatios = Table[
    {i, robinRatio[primorials[[i]]]}, 
    {i, 7, kMax}
  ];
  
  Print["Exp(gamma) limit: ", eGamma];
  Print["Max observed ratio: ", Max[robinRatios[[All, 2]]]];
  
  ListLinePlot[robinRatios, 
    AxesLabel -> {"k", "G(N_k)"},
    PlotLabel -> "Robin Ratio vs. Primorial Index",
    Epilog -> {Red, Dashed, InfiniteLine[{0, eGamma}, {1, 0}]}]
]

Conclusions

The analysis of hal-04454257v1 reinforces the deep connection between the discrete world of divisor sums and the continuous landscape of complex analysis. By refining the Robin and Nicolas criteria, the research provides a more precise mapping of how the distribution of prime numbers governs the upper bounds of arithmetic abundance. The most promising avenue for progress lies in the smallest counterexample theory, which uses colossally abundant numbers to pin down the exact oscillation required to break the Robin inequality. These findings suggest that the path to proving the Riemann Hypothesis may lie in the increasingly tight constraints we can place on the growth of arithmetic functions.

References

hal-04454257v1: On the zeros of the Riemann zeta function
Robin, G. (1984). Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann.
Nicolas, J. L. (1983). Petites valeurs de la fonction d'Euler.