Robin Criterion on Divisibility

Robin 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 $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. This is known as the Robin inequality. We prove that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. In addition, we demonstrate that the Robin inequality holds when $\frac{\pi^{2}}{6} \times \log\log n' \leq \log\log n$ for some number $n>5040$ where $n'$ is the square free kernel of the natural number $n$. Using these results, we show some properties of the possible smallest counterexample $n > 5040$ of the Robin inequality.