The Complete Proof of 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. We prove that the Robin inequality is true for all \(n > 5040\) which are not divisible by any prime number between $2$ and $953$. Using this result, we show there is a contradiction just assuming the possible smallest counterexample \(n > 5040\) of the Robin inequality. In this way, we prove that the Robin inequality is true for all \(n > 5040\) and thus, the Riemann Hypothesis is true.