The Riemann Hypothesis

EasyChair Preprint no. 3708, version 43

14 pagesDate: March 23, 2021

Abstract

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 $\frac{1}{2}$. 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. 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 prove the Robin's inequality is true for every integer $n > 5040$ that is not divisible by any prime $q_{m} \leq 47$. Besides, we demonstrate the Lagarias's inequality is true for every integer $n > 5040$ when $n = r \times q_{m}$ and the Lagarias's inequality is true for $r$, where $q_{m} \geq 47$ denotes the largest prime factor of $n$. We finally show the union of these results implies the proof of the Lagarias's inequality and therefore, the Riemann Hypothesis must be true.

Keyphrases: Divisor, inequality, number theory