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# A Proof of the Riemann Hypothesis Based on MacLaurin Expansion and Hadamard Product 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 and Hadamard Product of the Completed Zeta Function. Preprints 2021, 2021080146 (doi: 10.20944/preprints202108.0146.v10). Zhang, W. A Proof of the Riemann Hypothesis Based on MacLaurin Expansion and Hadamard Product of the Completed Zeta Function. Preprints 2021, 2021080146 (doi: 10.20944/preprints202108.0146.v10).

## Abstract

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) by conjugate complex roots $\alpha_i\pm j\beta_i, i\in \mathbb{N}$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}$$ which is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution $\alpha_i= \frac{1}{2}, i\in \mathbb{N}$. Therefore, 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

Comment 1
Commenter: Zhang Weicun
Commenter's Conflict of Interests: Author
Comment: The ABSTRACT and CONCLUSION are rewritten to be more readable. Two equations are added in the proof of Lemma 3.
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