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

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} < e^γ\sum_{j=0}^{ω(n)} \frac{(\ln\ln\ln n)^j}{j!}, \qqu