Download PDFOpen PDF in browserCurrent versionThe Riemann HypothesisEasyChair Preprint 3708, version 40Versions: 12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152→history 10 pages•Date: February 23, 2021AbstractIn 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 sumofdivisors function and $\gamma \approx 0.57721$ is the EulerMascheroni 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 $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $H_{n}$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a proof of the Riemann Hypothesis. Keyphrases: Divisor, inequality, number theory
