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. In 2007, Choie, Lichiardopol, Moree and Sol{\'{e}} have shown that the Robin inequality is true for all $n > 5040$ which are not divisible by $2$. We prove that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $3$ and $953$.