Properties of the Robin’s Inequality

EasyChair Preprint no. 3708, version 15

8 pagesDate: September 7, 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. The Robin's inequality consists in $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ where $\sigma(n)$ is the divisor 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 prove the Robin's inequality is true for every natural number $n > 5040$ when $n$ is not divisible by $3$. More precisely: every possible counterexample $n > 5040$ of the Robin's inequality must comply that $n$ should be divisible by $2^{20} \times 3^{13}$. In addition, the Robin's inequality is true for every natural number $n > 5040$ when $n = 3^{k} \times m$, $\frac{3}{2} \times \ln \ln m \leq \ln \ln (3^{k} \times m)$ and $3 \nmid m$. Moreover, we demonstrate the Robin's inequality is true for every natural number $n > 5040$ when $n$ is not divisible by $5$. Furthermore, we show the Robin's inequality is true for every natural number $n > 5040$ when $n$ is not divisible by $7$.

Keyphrases: Divisor, inequality, number theory