The Equivalence of Robin Inequality and the Riemann Hypothesis
The Riemann Hypothesis remains one of the most profound challenges in number theory. Recent research, particularly in papers like hal-03379098 and hal-03356311, leverages Robin inequality as a primary tool for investigation. This inequality states that for all natural numbers n greater than 5040, the sum-of-divisors function sigma(n) is bounded by e^gamma * n * log(log n), where gamma is the Euler-Mascheroni constant. It has been established that this inequality is equivalent to the Riemann Hypothesis.
Hardy-Ramanujan Integers and Superabundant Numbers
A central pillar of this approach is the study of Hardy-Ramanujan integers. These are numbers whose prime factorization exponents are non-increasing. Specifically, if n is the product of prime factors q_i raised to the power a_i, then a1 >= a2 >= ... >= am >= 0. Research suggests that if a counterexample to Robin inequality exists, the smallest such number must be a superabundant number, which inherently possesses the Hardy-Ramanujan structure.
The Yn Sequence and Chebyshev Bounds
To refine the bounds on the prime counting function and the Chebyshev function theta(x), researchers have introduced the sequence Yn = exp(1 / (2 * log(pn))) / (1 - 1 / log(pn)). This sequence acts as a convergence regulator. By analyzing the inequality log(Ym * theta(qm)) > log(qm) + log(Ym * (1 - 1 / log(qm))), we can better understand the oscillations of prime distributions. These oscillations are directly linked to the distribution of zeros in the Riemann zeta function.
The Role of the Squarefree Kernel and Zeta(2)
Another significant framework involves the squarefree kernel n-prime, which is the product of distinct primes dividing n. By using the identity (1 / (1 - q^-2)) * ((q + 1) / q) = q / (q - 1), we can establish a bridge between the behavior of general integers and their squarefree counterparts. This identity incorporates the value of the zeta function at s=2, which is pi^2 / 6. The research suggests that if the inflation caused by this factor is controlled, Robin inequality must hold for all n greater than 5040.
A Proposed Research Agenda for a Formal Proof
- Kernel-Case Classification: Divide the problem into cases where the squarefree kernel is either greater than or less than 5040. Use the growth gap log(log n-prime) vs log(log n) to prove the inequality for larger kernels.
- Exponent Constraints: For smaller kernels, utilize the non-increasing property of Hardy-Ramanujan exponents to show that the growth of log(log n) eventually dominates the sum-of-divisors ratio.
- Monotonicity Analysis: Prove that certain operators involving the Yn sequence are monotonically decreasing, thereby limiting the range of potential counterexamples to a finite, verifiable set.
By combining these frameworks, mathematicians can transform a complex analytic problem into a discrete optimization task, potentially closing the gap in our understanding of the Riemann zeta function.