Properties of the First Possible Counterexample in the Robin's Inequality

EasyChair Preprint no. 4481, version 2

Versions: 12history
4 pagesDate: October 29, 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. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ 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. This inequality is known today as the Robin's inequality. We demonstrate an interesting result about the smallest possible counterexample exceeding $5040$ of the Robin's inequality. The existence of such counterexample seems unlikely according to the evidence of this result. 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