Properties of the Robin’s Inequality

EasyChair Preprint no. 3708, version history

VersionDatePagesVersion notes
1July 1, 20204
2July 25, 20204

We change abstract, keywords and content.

3July 26, 20204

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

4July 27, 20204

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

5July 29, 20204

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.

6August 20, 20207

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

7August 25, 20208

We have fixed the title abstract, keywords and content.

8August 27, 20207

We have fixed the flaws.

9August 27, 20208

We have changed the title, the keywords and content.

10August 27, 20208

We have fixed few details.

11August 28, 20207

We have fixed the last details.

12August 30, 20206

We made the work as short and clear as possible.

13August 31, 20207

Fixed last flaw.

14September 1, 20207

Added missing details.

15September 7, 20208

We modify the abstract and content.

16September 8, 20209

We modify the abstract and content.

17September 10, 20209

Fixed minor details.

18September 13, 202011

Changed abstract and content.

19September 14, 202010

Minor changes: abstract and content.

20October 3, 202011

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

21October 4, 202011

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.

22October 9, 202010

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

23October 11, 20206

We make the arguments shorter.

24October 26, 20207

Minor changes.

25November 9, 202016

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.

26November 10, 202014

Small detail.

27November 16, 202012

Minor changes: abstract and content were changed.

28November 17, 202012

Minor changes.

29November 19, 202013

Minor change.

30November 19, 202012

Minor changes: abstract and content were changed.

31November 20, 202011

Minor changes: abstract and content were changed.

32November 25, 202010

Abstract and content were changed.

33November 30, 202012

Title, abstract and content were changed.

34December 2, 202010

Title, abstract and content were changed.

35December 7, 202010

Include it the Acknowledgments section.

36December 25, 202010

Removed the computational evidence as the principal argument.

37December 28, 202010

We specify the differentiation rule is in the variable r.

38February 5, 202110

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

39February 18, 202110

I have improved the last paragraph of the paper.

40February 23, 202110

Improved the Lemma 3.1: making easier to understand.

41March 18, 202111

We improve the abstract and another parts of the paper.

42March 22, 202114

We improved the theorem 1.6 and changed the abstract.

43March 23, 202114

Improved theorem 1.6 and put Acknowledgments section

44March 24, 202114

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

45March 30, 202114

We improve the Theorem 8.2

46April 27, 20214

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

47May 7, 20214

Improved version: abstract, keywords and content were changed

48May 10, 20215

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

49May 11, 20215

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

50June 15, 20215

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

51July 9, 20218

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

52July 15, 20216

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

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:
  author = {Frank Vega},
  title = {Properties of the Robin’s Inequality},
  howpublished = {EasyChair Preprint no. 3708},

  year = {EasyChair, 2021}}