On Robin's criterion for the Riemann Hypothesis

Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma\approx 0.57721$ is the Euler-Mascheroni constant. We call this inequality, the Robin inequality. In 2022, Vega stated that the possible existence of the smallest counterexample $n>5040$ of the Robin inequality implies that $q_{m} > e^{31.018189471}$ and $(\log n)^{\beta} < 1.03352795481\times\log(N_{m})$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$, $q_{m}$ is the largest prime divisor of $n$ and $\beta = \prod_{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$ when $n$ must be an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$. Based on that result, we obtain a contradiction just assuming the existence of such possible smallest counterexample $n > 5040$ for the Robin inequality. By contraposition, we show that the Riemann hypothesis should be true.