The Robin’s Inequality

We change abstract, keywords and content.

We have improved the arguments and precisely stated what was the computational evidence.

We increment the upper bound to $10^{1875}$.

We remove the computational evidence, since this was already proved in 2007: The result has been proved for all integers n not divisible by 2 (which are bigger than 10) It is Theorem 1.2 of  a paper of Choie, Lichiardopol, Moree and Sole. In this new version, we show if the Robin's inequality is not true for every natural number n > 5040, then this has an infinite number of counterexamples. However, the Robin's inequality cannot have an infinite number of counterexamples according to the asymptotic growth rate of the sigma function. Consequently, the Riemann hypothesis is true.

We fixed the arguments: We changed abstract, keywords and content.

We have fixed the title abstract, keywords and content.

We have fixed the flaws.

We have changed the title, the keywords and content.

We have fixed few details.

We have fixed the last details.

We made the work as short and clear as possible.

Fixed last flaw.

Added missing details.

We modify the abstract and content.

We modify the abstract and content.

Fixed minor details.

Changed abstract and content.

Minor changes: abstract and content.

We change the title, abstract, keywords and content just improving the paper.

Note the title was not changed from the web page in the last version even though I changed in the pdf paper (the new pdf paper and the whole metadata is still the same). This may not required a new version or a new pdf file, but simply change the title of the last version in the web page by the new title that has actually the pdf paper in the last version, that is: The Robin's Inequality.

We modifiy the content, abstract and title. This time we achieved to prove the Riemann Hypothesis.

We make the arguments shorter.

Minor changes.

Abstract, keywords and content were changed. Besides, this new version was supported by another researcher that shall be included as an author after his approval.

Small detail.

Minor changes: abstract and content were changed.

Minor changes.

Minor change.

Minor changes: abstract and content were changed.

Minor changes: abstract and content were changed.

Abstract and content were changed.

Title, abstract and content were changed.

Title, abstract and content were changed.

Include it the Acknowledgments section.

Removed the computational evidence as the principal argument.

We specify the differentiation rule is in the variable r.

Fixed the last page of the paper with a better explanation of the arguments

I have improved the last paragraph of the paper.

Improved the Lemma 3.1: making easier to understand.

We improve the abstract and another parts of the paper.

We improved the theorem 1.6 and changed the abstract.

Improved theorem 1.6 and put Acknowledgments section

Now, the content is better organized and the abstract is simpler.

We improve the Theorem 8.2

The abstract, keywords and content changed: Now the result is opposite from the older version.

Improved version: abstract, keywords and content were changed

We take into account the function S(x) could be negative.

We make the result for all number x >= 127 instead of the old result that was for all prime p >= 127.

We changed the Abstract, Keywords, References and Content of the paper.

We changed the abstract and content of the paper. This may require a new version.

We replaced the Robin inequality by Nicolas inequality to show a simpler result.