Investigating the Riemann Hypothesis via Robin and Nicolas Criteria

The Riemann Hypothesis and Arithmetic Criteria

The Riemann Hypothesis remains a central problem in number theory. As explored in arXiv 2111.06325, the hypothesis is equivalent to Robin's inequality, which involves the sum-of-divisors function σ(n) for integers greater than 5040.

Interplay of Robin and Nicolas Criteria

The research establishes a deep connection between Robin's inequality and the Nicolas criterion, which concerns the ratio of primorials to their Euler totient function value φ(n). The study focuses on Hardy-Ramanujan integers, which are numbers where the prime exponents are non-increasing.

• Extremal Candidates: Hardy-Ramanujan integers are shown to be the most likely candidates for violating Robin's inequality due to their high density of small prime factors.
• Mathematical Duality: If Robin's criterion fails for a specific integer n, it implies that the Nicolas criterion must hold for the corresponding primorial N_k composed of the same number of prime factors.
• Asymptotic Behavior: The study utilizes the properties of the Euler-Mascheroni constant γ to bound these arithmetic functions and test the limits of the hypothesis.

By synthesizing these two criteria, the paper provides a refined framework for investigating potential counterexamples to the Riemann Hypothesis through the lens of highly structured integer sequences and their divisor properties.