Robin's criterion on divisibility

Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n > 5040$, 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 $\frac{\pi^{2}}{6} \times\log\logn'\leq\log\logn$ 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 $q_{m}>e^{30.99733785}$, $1<\frac{1.25\times\log(4.7312714399)}{\log q_{m}}+\frac{\log N_{m}}{\log n}$, $(\log n)^{\beta}<1.0501395952\times\log(N_{m})$ and $n<(4.7312714399)^{m}\timesN_{m}$, 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 $\prod_{i=1}^{m}q_{i}^{a_{i}}$.