Riemann, Metatheory, and Proof, Rev.3

Riemann, Metatheory, and Proofs, 2020

Historical analysis and new concepts enable understanding of Riemann's Hypothesis (RH), his zeta function, mathematics, the principles enabling them, and ontological proofs. They enable new, post-modern metamathematics. New terms and definitions support comprehensive proof of new number theory, set theory, and proof theory. They enable deep understanding of the full scope of the historic context and causes of RH. The "Results" section provides basics and the technical context of work on RH and related problems. The "History" section summarizes the background of useful work and attempted solutions. Section 2 enables understanding of the work, enabling metalogic, and the domain of discourse. Section 3 provides comprehensively definitive, deeply explanatory proof of RH, and an integral critique of prior metatheory. It includes proofs of closely related problems, explanations, and how primal numbers may be rapidly, economically located. The new metatheorems of ontology, mathematics, numbers, sets, and proof also support new conjectures and possibilities. Disproof of the P/NP problem is included. Section 4 provides summary comments and predictions, for inspiration and verification.

The Riemann Hypothesis, one of the most enduring and significant unsolved problems in mathematics, concerns the distribution of non-trivial zeros of the Riemann zeta function. Despite extensive computational and theoretical efforts, its definitive proof remains elusive. This work introduces the Eusynthetic Principle, a novel framework that extends the classical principle of non-contradiction by emphasizing the critical role of perfect premises, order, and coherence. By applying the Eusynthetic Principle, this paper demonstrates how the Riemann Hypothesis emerges as a coherent and self-similar construct, where the alignment of zeros on the critical line becomes a logical necessity. Furthermore, the fractal nature of the zeta function's zeros is explored as a manifestation of universal self-similarity, reinforcing the hypothesis's validity. This study not only validates the Riemann Hypothesis but also invites a reevaluation of foundational axioms in mathematics, aiming to construct knowledge systems that are robust, coherent, and impervious to incomplete premises.

This document explores the application of the Eusyntesi Principle to some of the most significant challenges in mathematics, transforming them into resolved laws or frameworks. By focusing on the Goldbach Conjecture, the Riemann Hypothesis, and other Millennium Prize Problems, we demonstrate how the Eusyntesi Principle unifies complex mathematical phenomena under a coherent and self-similar structure. The Goldbach Conjecture, resolved through this principle, has evolved from a hypothesis into a universal law that governs the distribution of prime numbers. Similarly, the Eusyntesi Principle provides a systematic approach to the Millennium Prize Problems, including the Riemann Hypothesis and others, offering clarity and solutions to longstanding challenges. Key results emphasize the transformative nature of the Eusyntesi Principle, not only in mathematics but as a universal framework for understanding order, symmetry, and coherence across disciplines. This work also advocates for a reevaluation of validation methods in mathematics, urging a shift from traditional peer review to synthetic validation rooted in the intrinsic logic of mathematics. Documents referenced in this study are currently in Italian and will be made available in English to ensure accessibility to the international community.

In this document, by means of a novel system model and first order topological, algebraic and geometrical free-context formal language (NT-FS&L), first, we describe a new signature for a set of the natural numbers that is rooted in an intensional inductive de-embedding process of both, the tensorial identities of the known as “natural numbers”, and the abstract framework of theirs locus-positional based symbolic representations. Additionally, we describe that NT-FS&L is able to: i.- Embed the De Morgan´s Laws and the FOL-Peano´s Arithmetic Axiomatic. ii.- Provide new points of view and perspectives about the succession, precede and addition operations and of their abstract, topological, algebraic, analytic geometrical, computational and cognitive, formal representations. Second, by means of the inductive apparatus of NT-FS&L, we proof that the family of conjectures known as Glodbach’s holds entailment and truth when the reasoning starts from the consistent and finitary axiomatic system herein described.

In this talk, we give a panorama of the proof of the Riemann Hypothesis we have in recent years, see [?], which is based on the proofs of the strong density hypothesis in [?] and the strong Lindelöf hypothesis in [?]. The proof of RH is "short", but the preparation is quite a lengthy work. For a rough sketch of the proof of the Riemann Hypothesis in a short article instead of a panorama as in this talk, one may see [?]. We shall discuss five symmetries: the conjugate symmetry, algebra and analysis or the prime counting function and the zero-free region of the Riemann zeta function, the pseudogamma functions, various alternation of the Riemann zeta functions and the various prime counting functions like π(x), ψ(x), and ϖ(x), the almost symmetric routes, and, the Lambda function with log(p) when n is a prime powers vesus −1 otherwise in five subjects: applied mathematics, algebra, analysis, computer algebra, and topology.

2018

The aim of this study is to prove Goldbach's famous Conjectures known as strong and weak conjectures. Strong Conjecture: "Every even number greater than 2 is the sum of two prime numbers". Weak Conjecture: "Every odd integer greater than 5 can be written as the sum of three prime numbers". Content: Searching prime numbers with predictive formulas is beyond the scope of this work. The program that generates and tests the prime number is a separate study, in this study the proofs of the Strong and Weak Goldbach Conjectures will take place. We only approach Gold-bach's Conjecture, where all required prior knowledge on prime numbers assumed as accepted by Goldbach's works. Therefore, we start with take a look at Goldbach's description of original problem then will try to derive a step by step proof upon that is described as in original letters. Method: Once the sub groups of the set of prime numbers were defined, theoretical framework was proved to be complete. The theoretical framework is very simple and concise, albeit the entire study is based upon that. This study is providing proof on both of conjectures. Findings and Results: In this study, an effective solution to a historical problem known in mathematics but not proven to this day is introduced; the proofs of Goldbach's Conjectures (both in Strong and Weak) are given. These proofs will open many obstacles in number theory and provide a fresh look on prime numbers and their applications. Many assumptions, conjectures in number theory will be re-evaluated. Based on this proof, another theorem on prime number is constructed with its proof. Conclusion: Many assumptions, conjectures about prime numbers will be re-evaluated under the light of given proofs. There is no reason to limit the sum in the theorem above; one can easily say that prime numbers are infinite. As a discussion, albeit there are institutionalized methods on computing prime numbers, a new way of computing bigger prime numbers can be based on this new perspective this paper has shed lights on. Proofs presented will introduce new horizons to relevant academicians on number theory. Defining this new perspective might also help one to expend this study further points related to even if there is a pattern on Prime Numbers so we can exploit that to compute bigger numbers feasibly. The prominence of major prime numbers in cryp-tology is known. Encryption will be redesigned in line with the proofs. The proof will open new horizons on prime numbers, the first one is this explained and proved new Prime Number Theorem: Aksoy Theorem.

More obstacles to a proof in rational arithmetic are discussed

In this study, the Riemann problem is presented with highlights on history of the zeta function.

Mathematics and Statistics, 2021

In 1859, Bernhard Riemann, a German mathematician, published a paper to the Berlin Academy that would change mathematics forever. The mystery of prime numbers was the focus. At the core of the presentation was indeed a concept that had not yet been proven by Riemann, one that to this day baffles mathematicians. The way we do business could have been changed if the Riemann hypothesis holds true, which is because prime numbers are the key element for banking and e-commerce security. It will also have a significant influence, impacting quantum mechanics, chaos theory, and the future of computation, on the cutting edge of science. In this article, we look at some well-known results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. Initially, we observe omitting a logical undefined term in the integral representation of Zeta function by the means of Gamma function. For that we propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: = 1 2 by assuming that () and (1 −) simultaneously equal zero. Consequently, we conditionally prove Riemann Hypothesis. MSC 2010 Classification: 97I80, 11M41

MaktuB Studio, 2024

For every prime number defined as "divisible by one and itself" I utilize Teotl Mathematics to study the behavior of such numbers and to find the system of complex functions that governs their behavior over time. Thereby, I present proof by construction that such system of equations exists. The analysis and distribution to be presented in this paper, are proof that; there is a complex system of functions and conditions that lead to the exact distribution, location and value of all prime numbers; hence, the sequence of prime numbers is predictable for all odd numbers, n.odd, along the critical line of prime roots with shared a ½+it value in Global Time. Furthermore, the Teotl method confirms the validity of the Riemann Hypothesis for the positive domain of integers along the critical line.