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# A Proof of 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 Riemann Hypothesis Based on MacLaurin Expansion of the Completed Zeta Function. Preprints 2021, 2021080146 (doi: 10.20944/preprints202108.0146.v1). Zhang, W. A Proof of Riemann Hypothesis Based on MacLaurin Expansion of the Completed Zeta Function. Preprints 2021, 2021080146 (doi: 10.20944/preprints202108.0146.v1).

## Abstract

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series. Thus, $\xi(s)=0$ corresponds to an algebraic equation with real coefficients and infinite degree. In addition, by $\xi(s)=\xi(1-s)$, another formally equivalent algebraic equation exists, i.e., $\xi(1-s)=0$. Then these two simultaneous algebraic equations share the common solution, thus a proof of Riemann Hypothesis (RH) can be obtained.

## Keywords

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

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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