The Riemann Hypothesis

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 $\frac{1}{2}$. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma (n)<e^{\gamma }\times n\times \log \log n$ holds for all sufficiently large $n$, where $\sigma (n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n>5040$ if and only if the Riemann Hypothesis is true. In 2002, Lagarias proved that if the inequality $\sigma (n)\le H_n+exp(H_n)\times \log H_n$ holds for all $n\ge 1$, then the Riemann Hypothesis is true, where $H_n$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a proof of the Riemann Hypothesis.