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Properties of the Robin’s Inequality

EasyChair Preprint no. 3708, version 19

10 pagesDate: September 14, 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 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

BibTeX entry
BibTeX does not have the right entry for preprints. This is a hack for producing the correct reference:
@Booklet{EasyChair:3708,
  author = {Frank Vega},
  title = {Properties of the Robin’s Inequality},
  howpublished = {EasyChair Preprint no. 3708},

  year = {EasyChair, 2020}}
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