Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed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

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)
Version 4 : Received: 23 August 2021 / Approved: 24 August 2021 / Online: 24 August 2021 (11:48:54 CEST)
Version 5 : Received: 27 August 2021 / Approved: 30 August 2021 / Online: 30 August 2021 (10:33:44 CEST)
Version 6 : Received: 20 September 2021 / Approved: 22 September 2021 / Online: 22 September 2021 (10:16:56 CEST)
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)
Version 9 : Received: 11 October 2021 / Approved: 14 October 2021 / Online: 14 October 2021 (14:14:19 CEST)
Version 10 : Received: 23 October 2021 / Approved: 25 October 2021 / Online: 25 October 2021 (13:31:43 CEST)
Version 11 : Received: 11 November 2021 / Approved: 12 November 2021 / Online: 12 November 2021 (14:54:48 CET)
Version 12 : Received: 26 November 2021 / Approved: 26 November 2021 / Online: 26 November 2021 (10:15:27 CET)

How to cite: Zhang, W. A Proof of the Riemann Hypothesis based on MacLaurin Expansion of the Completed Zeta Function. Preprints 2021, 2021080146 (doi: 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 (doi: 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

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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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