Short Note about the Robin's Inequality

EasyChair Preprint no. 4169, version 3

Versions: 123history
4 pagesDate: October 28, 2020

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 $\frac{1}{2}$. 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. The Robin's inequality consists in $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Robin's inequality is true for every natural number $n > 5040$ if and only if the Riemann Hypothesis is true. We demonstrate an interesting result about the smallest possible counterexample of the Robin's inequality exceeding $5040$. However, according to this result, the existence of such counterexample seems unlikely. In this way, we provide a new step forward in the efforts of trying to prove the Riemann Hypothesis.

Keyphrases: Divisor, inequality, number theory, Prime