Robin's criterion on divisibility

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 show that the Robin inequality is true for all natural numbers $n>5040$ that are not divisible by some prime between $2$ and $1771559$. We prove that the Robin inequality holds when $\frac{\pi^{2}}{6}\times\log\log n' \leq\log\log n$ for some $n>5040$ where $n'$ is the square free kernel of the natural number $n$. The possible smallest counterexample $n>5040$ of the Robin inequality implies that $q_{m}>e^{31.018189471}$, $1<\frac{(1+\frac{1.2762}{\log q_{m}})\times \log(2.82915040011)}{\log\log n}+\frac{\log N_{m}}{\log n}$, $(\log n)^{\beta}<1.03352795481\times\log(N_{m})$ and $n<(2.82915040011)^{m}\times N_{m}$, 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 $n$ is an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m}q_{i}^{a_{i}}$.