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 2002, Lagarias proved that if the inequality $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $\sigma(n)$ is the sum-of-divisors function and $H_{n}$ is the $n^{th}$ harmonic number. We prove this inequality holds for all $n \geq 1$ and therefore, the Riemann Hypothesis must be true.