Version 1
: Received: 27 September 2024 / Approved: 27 September 2024 / Online: 30 September 2024 (09:28:49 CEST)
How to cite:
Oliveira, H. Investigating the Riemann Hypothesis with HQF ASA and Modular Surfaces - a Detailed Description. Preprints2024, 2024092267. https://doi.org/10.20944/preprints202409.2267.v1
Oliveira, H. Investigating the Riemann Hypothesis with HQF ASA and Modular Surfaces - a Detailed Description. Preprints 2024, 2024092267. https://doi.org/10.20944/preprints202409.2267.v1
Oliveira, H. Investigating the Riemann Hypothesis with HQF ASA and Modular Surfaces - a Detailed Description. Preprints2024, 2024092267. https://doi.org/10.20944/preprints202409.2267.v1
APA Style
Oliveira, H. (2024). Investigating the Riemann Hypothesis with HQF ASA and Modular Surfaces - a Detailed Description. Preprints. https://doi.org/10.20944/preprints202409.2267.v1
Chicago/Turabian Style
Oliveira, H. 2024 "Investigating the Riemann Hypothesis with HQF ASA and Modular Surfaces - a Detailed Description" Preprints. https://doi.org/10.20944/preprints202409.2267.v1
Abstract
This article describes a new application of the global optimization method HQF ASA to the investigation of the Riemann hypothesis, in the sense of searching for zeros of the Riemann zeta function ζ outside the critical line but inside the (open) subset of the critical strip ((0,1/2)∪(1/2,1))×(−∞,∞). The underlying idea is very simple: a complex root r of a given complex function f : C → C must satisfy the condition ||f(r)|| = 0, that is, f(r) must have 2-dimensional Eu- clidean norm equal to 0, considering r as a point of R2. In the opposite direction, for any r ∈ C, ||f(r)|| = 0 implies f(r) = 0 and r is a root of f. Focusing again on the Riemann zeta function ζ, the open and simply connected region Rc = (1/2,1)×(0,∞) is used along the text, taking into account the symmetries of zeros of ζ in the critical strip. In this fashion, finding a global optimizer for ||ζ|| is equivalent to finding a root of ζ. Therefore, and taking into account the well-known efficacy of HQF ASA, a good sign of the trueness of RH would be not finding global minima in several optimization sessions with expanding subsets of Rc. In practical terms, several optimization sessions are executed using QMF ASA with increasing subsets of Rc, searching for global minima of ||ζ||. After presenting numerical results and figures, the adequacy of the proposed approach as a tool for preliminary empirical analysis for this type of problem is discussed.
Keywords
Riemann Hypothesis;Riemann Zeta Function;Artificial Inference;HQF ASA; Global Optimization.
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.
