A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function

Weicun Zhang

doi:10.20944/preprints202108.0146.v4

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preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202108.0146.v4

Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function

Weicun Zhang

Version 1 : Received: 2 August 2021 / Approved: 5 August 2021 / Online: 5 August 2021 (12:21:56 CEST)
Version 2 : Received: 6 August 2021 / Approved: 9 August 2021 / Online: 9 August 2021 (12:41:53 CEST)
Version 3 : Received: 20 August 2021 / Approved: 20 August 2021 / Online: 20 August 2021 (11:52:39 CEST)
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Version 7 : Received: 24 September 2021 / Approved: 27 September 2021 / Online: 27 September 2021 (12:22:55 CEST)
Version 8 : Received: 4 October 2021 / Approved: 5 October 2021 / Online: 5 October 2021 (12:40:07 CEST)
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How to cite: Zhang, W. A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function. Preprints 2021, 2021080146. https://doi.org/10.20944/preprints202108.0146.v4 Zhang, W. A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function. Preprints 2021, 2021080146. https://doi.org/10.20944/preprints202108.0146.v4

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MDPI and ACS Style

Zhang, W. A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function. Preprints 2021, 2021080146. https://doi.org/10.20944/preprints202108.0146.v4

APA Style

Zhang, W. (2021). A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function. Preprints. https://doi.org/10.20944/preprints202108.0146.v4

Chicago/Turabian Style

Zhang, W. 2021 "A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function" Preprints. https://doi.org/10.20944/preprints202108.0146.v4

Abstract

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series (infinite polynomial). Thus, by $\xi(s)=\xi(1-s)=0$, we have the following infinite polynomial equation \begin{equation}\nonumber \begin {split} &\xi(0)+\xi^{'}(0)s+\frac{\xi^{''}(0)}{2!}s^{2}+\cdots+\frac{\xi^{(n)}(0)}{n!}s^{n}+\cdots\\ =&\xi(0)+\xi^{'}(0)(1-s)+\frac{\xi^{''}(0)}{2!}(1-s)^{2}+\cdots+\frac{\xi^{(n)}(0)}{n!}(1-s)^{n}+\cdots\\ =&0 \end {split} \end{equation} which finally leads to $s=1-s, s=\alpha \pm j\beta, \beta\neq 0$, then a proof of the Riemann Hypothesis can be achieved.

Keywords

Riemann Hypothesis (RH); Proof ; Completed zeta function $\xi(s)$

Subject

Computer Science and Mathematics, Algebra and 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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Comments (1)

Comment 1

Received: 24 August 2021
Commenter: Zhang Weicun
Commenter's Conflict of Interests: Author

Comment: With this updated version, the proof details of RH have been further improved( from four steps to three steps).

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