Version 1
: Received: 6 September 2023 / Approved: 7 September 2023 / Online: 7 September 2023 (11:19:39 CEST)
Version 2
: Received: 10 October 2023 / Approved: 10 October 2023 / Online: 10 October 2023 (10:13:06 CEST)
Version 3
: Received: 12 November 2023 / Approved: 13 November 2023 / Online: 13 November 2023 (10:09:02 CET)
Version 4
: Received: 19 November 2023 / Approved: 20 November 2023 / Online: 20 November 2023 (07:30:41 CET)
Version 5
: Received: 16 February 2024 / Approved: 21 February 2024 / Online: 21 February 2024 (04:40:09 CET)
How to cite:
Oliveira, H. A Modest Contribution to the Riemann Hypothesis Using the Poincaré Index. Preprints2023, 2023090513. https://doi.org/10.20944/preprints202309.0513.v5
Oliveira, H. A Modest Contribution to the Riemann Hypothesis Using the Poincaré Index. Preprints 2023, 2023090513. https://doi.org/10.20944/preprints202309.0513.v5
Oliveira, H. A Modest Contribution to the Riemann Hypothesis Using the Poincaré Index. Preprints2023, 2023090513. https://doi.org/10.20944/preprints202309.0513.v5
APA Style
Oliveira, H. (2024). A Modest Contribution to the Riemann Hypothesis Using the Poincaré Index. Preprints. https://doi.org/10.20944/preprints202309.0513.v5
Chicago/Turabian Style
Oliveira, H. 2024 "A Modest Contribution to the Riemann Hypothesis Using the Poincaré Index" Preprints. https://doi.org/10.20944/preprints202309.0513.v5
Abstract
This work uses a new approach to investigate the Riemann hypothesis, drawing conclusions about its trueness. It is based on the more general method presented in a very recent publication, by the same author, showing in detail how to approach that very well-known and interesting problem. This is achieved by means of the theory of dynamical systems (Poincaré index associated to equilibria of 2-dimensional systems) and the study of the zeros of the Dirichlet eta function, defined by a Dirichlet series. By using the well-known fact that the zeros of the eta function include all zeros of the Riemann zeta function in the (open) critical strip, excluded the critical line, ((0,1/2) ∪ (1/2,1)) × (−∞,∞), the development proceeds using only the eta function. In addition, the open and simply connected region (1/2,1) × (0,∞) is used along the text, taking into account the symmetries of zeros of the functions under analysis in the critical strip. The basic line of proof is to find the mathematical expression for the Poincaré index of the vector field associated to the eta function, assuming the existence of a zero of the eta function outside the critical line (in (1/2,1) × (0,∞)), and investigating the resulting unfoldings. Eventually, an inconsistency occurs and the proof ends by contradiction.
Keywords
Riemann hypothesis; Riemann zeta function; Poincaré index; Dirichlet eta function.
Subject
Computer Science and Mathematics, Mathematics
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.
