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# The Elementary Proof of the Riemann's Hypothesis

Version 1
: Received: 29 June 2020 / Approved: 30 June 2020 / Online: 30 June 2020 (10:30:45 CEST)

Version 2 : Received: 5 July 2021 / Approved: 6 July 2021 / Online: 6 July 2021 (11:32:03 CEST)

Version 2 : Received: 5 July 2021 / Approved: 6 July 2021 / Online: 6 July 2021 (11:32:03 CEST)

How to cite:
Feliksiak, J. The Elementary Proof of the Riemann's Hypothesis. *Preprints* **2020**, 2020060365 (doi: 10.20944/preprints202006.0365.v2).
Feliksiak, J. The Elementary Proof of the Riemann's Hypothesis. Preprints 2020, 2020060365 (doi: 10.20944/preprints202006.0365.v2).

## Abstract

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

## Keywords

Cramer`s conjecture; distribution of primes; elementary proof of the Riemann's Hypothesis; Landau problems; Legendre conjecture; Littlewood`s proof of 1914; logarithmic integral; maximal prime gaps; Prime Number Theorem; Tailored logarithmic integral; prime counting function Supremum; prime counting function Infimum.

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Commenter: Jan Feliksiak

Commenter's Conflict of Interests: Author

the prime counting function Infimum

the prime counting function lower bound.

In addition to that, a redundant definition had been removed and the maximal gaps proof had been slightly revised.