Properties of the Robin's Inequality

The Riemann hypothesis is considered the most important unsolved problem in mathematics. The Robin's inequality is true for every natural number $n > 5040$ if and only if the Riemann hypothesis is true. We prove the Robin's inequality is true for every natural number $n > 5040$ when $n$ is not divisible by any prime number $q_{m} \leq 113$. In addition, the Robin's inequality is true for every natural number $n = 113^{k} \times n' > 5040$ when $(\ln n')^{\beta} \leq \ln n$ such that $\beta = \frac{113}{112}$, $113 \nmid n'$ and for an integer $k \geq 1$. Moreover, given a natural number $n = q_{1}^{a_{1}} \times q_{2}^{a_{2}} \times \cdots \times q_{m}^{a_{m}}$ such that $n > 5040$, $q_{1}, q_{2}, \cdots, q_{m}$ are prime numbers and $a_{1}, a_{2}, \cdots, a_{m}$ are positive integers, then the Robin's inequality is true for $n$ when $q_{1}^{\alpha} \times q_{2}^{\alpha} \times \cdots \times q_{m}^{\alpha} \leq n$, where $\alpha = (\ln n')^{\beta}$, $\beta = (\frac{\pi^{2}}{6} - 1)$ and $n'$ is the squarefree kernel of $n$.