The Robin’s Inequality

EasyChair Preprint 3708, version history

VersionDatePagesVersion notes
1
July 1, 2020
4
2
July 25, 2020
4

We change abstract, keywords and content.

3
July 26, 2020
4

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

4
July 27, 2020
4

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

5
July 29, 2020
4

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.

6
August 20, 2020
7

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

7
August 25, 2020
8

We have fixed the title abstract, keywords and content.

8
August 27, 2020
7

We have fixed the flaws.

9
August 27, 2020
8

We have changed the title, the keywords and content.

10
August 27, 2020
8

We have fixed few details.

11
August 28, 2020
7

We have fixed the last details.

12
August 30, 2020
6

We made the work as short and clear as possible.

13
August 31, 2020
7

Fixed last flaw.

14
September 1, 2020
7

Added missing details.

15
September 7, 2020
8

We modify the abstract and content.

16
September 8, 2020
9

We modify the abstract and content.

17
September 10, 2020
9

Fixed minor details.

18
September 13, 2020
11

Changed abstract and content.

19
September 14, 2020
10

Minor changes: abstract and content.

20
October 3, 2020
11

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

21
October 4, 2020
11

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.

22
October 9, 2020
10

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

23
October 11, 2020
6

We make the arguments shorter.

24
October 26, 2020
7

Minor changes.

25
November 9, 2020
16

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.

26
November 10, 2020
14

Small detail.

27
November 16, 2020
12

Minor changes: abstract and content were changed.

28
November 17, 2020
12

Minor changes.

29
November 19, 2020
13

Minor change.

30
November 19, 2020
12

Minor changes: abstract and content were changed.

31
November 20, 2020
11

Minor changes: abstract and content were changed.

32
November 25, 2020
10

Abstract and content were changed.

33
November 30, 2020
12

Title, abstract and content were changed.

34
December 2, 2020
10

Title, abstract and content were changed.

35
December 7, 2020
10

Include it the Acknowledgments section.

36
December 25, 2020
10

Removed the computational evidence as the principal argument.

37
December 28, 2020
10

We specify the differentiation rule is in the variable r.

38
February 5, 2021
10

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

39
February 18, 2021
10

I have improved the last paragraph of the paper.

40
February 23, 2021
10

Improved the Lemma 3.1: making easier to understand.

41
March 18, 2021
11

We improve the abstract and another parts of the paper.

42
March 22, 2021
14

We improved the theorem 1.6 and changed the abstract.

43
March 23, 2021
14

Improved theorem 1.6 and put Acknowledgments section

44
March 24, 2021
14

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

45
March 30, 2021
14

We improve the Theorem 8.2

46
April 27, 2021
4

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

47
May 7, 2021
4

Improved version: abstract, keywords and content were changed

48
May 10, 2021
5

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

49
May 11, 2021
5

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

50
June 15, 2021
5

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

51
July 9, 2021
8

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

52
July 15, 2021
6

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