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 $\frac{1}{2}$. 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. 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. Given a natural number $n > 5040$, then the Robin inequality is true for $n$ when $\frac{\pi^{2}}{6} \times \ln \ln n' \leq \ln \ln n$ where $n'$ is the squarefree kernel of $n$. Moreover, we prove the Robin inequality is true for every natural number $n > 5040$ when $n$ is not divisible by any prime number $q_{m} \leq 113$. 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 $H_{n}$ is the $n^{th}$ harmonic number. Furthermore, let $n > 5040$ and $n = r \times q$, where $q$ denotes the largest prime factor of $n$: We demonstrate that if the Robin inequality is satisfied in $r$, then the Lagarias inequality is satisfied in $n$, when $q$ is a sufficiently large number.