A finite arithmetic form of Robin's inequality and its equivalence to the Riemann hypothesis

Source abstract

Robin's inequality reformulates the Riemann hypothesis as a bound on the sum-of-divisors function $σ$. We introduce a finite-truncation version of Robin's inequality determined intrinsically by the arithmetic structure of $n$, $$\frac{σ(n)}{n} 5040,$$ where $ω(n)$ denotes the number of distinct prime factors of $n$, and $γ$ is the Euler-Mascheroni constant. The resulting bound is pointwise stronger than Robin's inequality and we prove it unconditionally for integers with $ω(n)\le 6$, primorials, odd integers and square-free integers. We also prove that this inequality is equivalent to Robin's inequality, and hence to the Riemann hypothesis. Consequently, the Riemann hypothesis is equivalent to the truncated inequality holding for all colossally abundant numbers. We further show that if the inequality fails, its minimal counterexample must be a superabundant number, placing any obstruction within a rigid extremal class of integers.