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 show that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. In addition, from the possible smallest counterexample $n > 5040$ of the Robin inequality, we prove that necessarily $(\log n)^{\beta} < 1.26 \times \log(N_{m})$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$ 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 $\prod_{i=1}^{m} q_{i}^{a_{i}}$.