Properties of the Robin's Inequality

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. Many consider it to be the most important unsolved problem in pure mathematics. The Robin's inequality consists in $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ where $\sigma(n)$ is the divisor function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. 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 $n = 113^{k} \times n' > 5040$ over an integer $k \geq 1$ when $(\ln n')^{\beta} \leq \ln n$, such that $\beta = \frac{113}{112}$ and $n'$ is not divisible by $113$.