Preprint Hypothesis Version 2 Preserved in Portico This version is not peer-reviewed

# On the Riemann Hypothesis for the Zeta Function

Version 1 : Received: 24 May 2021 / Approved: 25 May 2021 / Online: 25 May 2021 (14:44:37 CEST)
Version 2 : Received: 25 May 2021 / Approved: 26 May 2021 / Online: 26 May 2021 (11:44:31 CEST)
Version 3 : Received: 31 May 2021 / Approved: 1 June 2021 / Online: 1 June 2021 (07:58:45 CEST)

How to cite: Yang, X. On the Riemann Hypothesis for the Zeta Function. Preprints 2021, 2021050616 (doi: 10.20944/preprints202105.0616.v2). Yang, X. On the Riemann Hypothesis for the Zeta Function. Preprints 2021, 2021050616 (doi: 10.20944/preprints202105.0616.v2).

## Abstract

In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

## Subject Areas

Riemann zeta--function; entire Riemann zeta--function; completed Riemann zeta--function; Riemann $\Xi$--function; Riemann hypothesis

Comment 1
Commenter: Xiao-Jun Yang
Commenter's Conflict of Interests: Author
Comment: We only revised the Remark in Section 4.
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