Properties of the Robin’s Inequality

EasyChair Preprint no. 3708, version 20

11 pagesDate: October 3, 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 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. Given a natural number $n = q_{1}^{a_{1}} \times q_{2}^{a_{2}} \times \cdots \times q_{m}^{a_{m}} > 5040$ such that $q_{1}, q_{2}, \cdots, q_{m}$ are prime numbers and $a_{1}, a_{2}, \cdots, a_{m}$ are natural numbers, then the Robin's inequality is true for $n$ when $q_{1}^{\alpha} \times q_{2}^{\alpha} \times \cdots \times q_{m}^{\alpha} \leq n$, where $\alpha = (\ln n')^{\beta}$, $\beta = (\frac{\pi^{2}}{6} - 1)$ and $n'$ is the squarefree kernel of $n$. Moreover, we prove the Robin's inequality is true for every natural number $n > 5040$ when $n$ is not divisible by any prime number $q_{m} \leq 113$. In addition, the Robin's inequality is true for every natural number $n = 113^{k} \times n' > 5040$ over an integer $k \geq 1$ when $(\ln n')^{\beta} \leq \ln n$, such that $\beta = \frac{113}{112}$ and $n'$ is not divisible by $113$.

Keyphrases: Divisor, inequality, number theory