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. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. The Robin's inequality consists in $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ where $\sigma(n)$ is the sum-of-divisors 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 $n > 5040$. In this way, we demonstrate the Riemann Hypothesis is true.

First Author: This version is in case something unpredictable could happen to me due to my current situation. Second Author: this author can claim the ownership of this paper whenever he wish. This version contains some flaws introduced only by me.