Properties of the Robin’s Inequality

EasyChair Preprint 3708, version 18

Abstract

The Riemann hypothesis is considered the most important unsolved problem in mathematics. 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 any prime number $q_{m} \leq 113$. In addition, the Robin's inequality is true for every natural number $n = 113^{k} \times n' > 5040$ when $(\ln n')^{\beta} \leq \ln n$ such that $\beta = \frac{113}{112}$, $113 \nmid n'$ and for an integer $k \geq 1$. Moreover, given a natural number $n = q_{1}^{a_{1}} \times q_{2}^{a_{2}} \times \cdots \times q_{m}^{a_{m}}$ such that $n > 5040$, $q_{1}, q_{2}, \cdots, q_{m}$ are prime numbers and $a_{1}, a_{2}, \cdots, a_{m}$ are positive integers, 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 $n$.

Keyphrases: Divisor, inequality, number theory

@booklet{EasyChair:3708,
  author    = {Frank Vega},
  title     = {Properties of the Robin’s Inequality},
  howpublished = {EasyChair Preprint 3708},
  year      = {EasyChair, 2020}}