The Riemann Hypothesis

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 obtain a contradiction just assuming the smallest counterexample of the Robin inequality exists for some $n > 5040$. In this way, we prove that the Robin inequality is true for all $n > 5040$. Consequently, the Riemann Hypothesis is also true.