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 $15 \nmid n$, where $15 \nmid n$ means that $n$ is not divisible by $15$. More specifically: every counterexample should be divisible by $2^{20} \times 3^{13} \times 5^{8} \times k_{1}$ or either $2^{20} \times 3^{13} \times k_{2}$ or $2^{20} \times 5^{8} \times k_{3}$, where $k_{1}, k_{2}, k_{3} > 1$, $2 \nmid k_{1}$, $3 \nmid k_{1}$, $5 \nmid k_{1}$, $2 \nmid k_{2}$, $3 \nmid k_{2}$, $2 \nmid k_{3}$ and $5 \nmid k_{3}$.