It is shown that there is a direct connection between the Riemann zeros, the counting numbers, and hence the prime numbers, but not in the so-called Critical Strip.
A mathematical structure is developed in which the articulation of the numerical location of a Riemann zero in the sequence of zeros is sufficient to determine the counting number with which it is associated, its location, Im(s) on the line of symmetry and, the Gauss/Legendre prime number counting function.
It is concluded that the disposition of the prime numbers within the system of counting numbers is solely an intrinsic characteristic of that system and is totally unrelated to the distribution of the Riemann zeros.
Table of Contents
Abstract
Introduction
Analysis
The assigning of values of the prime number counting function to the Riemann zeros along the line passing through x = ½
The impossibility of an ordinate of any Riemann zero in the Critical Strip to be directly associated with an integer
A critique of the importance attributed to the Critical Strip
Discussion and conclusion
Research Objectives and Core Themes
This work aims to challenge the traditional mathematical focus on the Riemann Critical Strip by demonstrating that the distribution of prime numbers is inherently linked to the counting number system itself, rather than to the zeros of the Riemann zeta function within the Critical Strip. The research establishes a direct connection between specific Riemann zeros and counting numbers along alternative ordinate lines, suggesting that the "Critical Strip" is not the optimal location for understanding prime number distribution.
- The relationship between the Riemann zeta function and counting numbers.
- Critique of the significance attributed to the Critical Strip.
- The concept of "Dirichlet lines" and their role in identifying prime numbers.
- Derivation of the Gauss/Legendre prime counting function from Riemann zero locations.
- Proof of the impossibility of associating Critical Strip ordinates with integers.
Excerpt from the Book
A critique of the importance attributed to the Critical Strip
Consider an abstract of the graph of the prime number counting functions against the nearodds, as shown in Fig3.
The graph of the prime number counting function must only consist of a series of horizontal lines spaced apart in the vertical direction by one unit. In the diagram a and b are taken to be prime numbers, as are c and e, and f and g, with b = c and e = f.
There can be no connection of any kind between b and c for example, for non-integer values of the prime number function have no meaning, and, if a and b are the only prime numbers on the line ab, then any other numbers, even, odd, and non-integer in the interval between a and b contribute nothing to the magnitude of the prime number counting function.
Summary of Chapters
Abstract: Presents the author's argument that the Riemann zeros within the Critical Strip are irrelevant to the distribution of primes, proposing instead a direct connection between the system of counting numbers and specific Riemann zeros.
Introduction: Provides a historical overview of Riemann's 1859 paper and outlines the long-standing mathematical fixation on the Critical Strip, which the author argues is largely unproductive.
Analysis: Details the mathematical framework and derivations, specifically utilizing Riemann's functional equation and Dirichlet functions to relate the zeta function to the counting numbers.
The assigning of values of the prime number counting function to the Riemann zeros along the line passing through x = ½: Demonstrates how an iterative process can link Riemann zeros to prime numbers, effectively "bracketing" them within the matrix of numbers.
The impossibility of an ordinate of any Riemann zero in the Critical Strip to be directly associated with an integer: Provides a mathematical argument showing that zeros in the Critical Strip cannot be linked to integers, thereby limiting their utility for counting prime numbers.
A critique of the importance attributed to the Critical Strip: Offers a conceptual evaluation of the graph of prime counting functions, reinforcing the argument that the Critical Strip does not provide a meaningful connection to prime number distribution.
Discussion and conclusion: Synthesizes the findings, concluding that prime number disposition is an intrinsic characteristic of the counting number system and independent of the Riemann zero distribution in the Critical Strip.
Keywords
Riemann zeros, Riemann zeta function, Critical Strip, Prime number counting function, Dirichlet line, Gauss/Legendre prime number counting function, Number theory, Counting numbers, Nearodd, Mathematical analysis, Riemann conjecture, Trivial zeros, Prime numbers, Complex plane, Ordinates
Frequently Asked Questions
What is the core argument of this publication?
The author argues that the standard focus on the Riemann Critical Strip is misplaced, proposing that the distribution of prime numbers is an intrinsic property of the counting numbers themselves, unrelated to the zeros found in the Critical Strip.
Which mathematical tool is primarily used in this research?
The author employs Riemann's functional equation and its extension through analytic continuation, alongside the development of the "Dirichlet line" concept to map Riemann zeros to counting numbers.
What is the author's view on the Critical Strip?
The author regards the Critical Strip as a "mathematical graveyard" that provides little actual insight into prime distribution, arguing that it is difficult to determine zero positions there with precision.
How are prime numbers linked to Riemann zeros in this work?
The author establishes a mapping where specific Riemann zeros on ordinates passing through trivial zeros correspond directly to counting numbers, which in turn are linked to the prime number counting function.
What is a "nearodd" as defined in the text?
A "nearodd" is a term coined by the author to denote an odd number associated with a specific Riemann zero in the matrix, used to calculate the magnitude of the prime number counting function.
What does the author suggest for future mathematical research?
The author concludes that researchers should shift their attention away from the Critical Strip and instead focus on ordinates passing through the trivial zeros of the Riemann zeta function to find a meaningful link to prime numbers.
Why can't Riemann zeros in the Critical Strip be associated with integers?
The author demonstrates that if such an association were possible, it would imply the zeros lie on Dirichlet lines; however, since these zeros do not contribute to the prime counting function, such an association remains impossible.
How does the author utilize Table 1 in the study?
Table 1 serves as a matrix of odd numbers that allows for the identification and location of Riemann zeros, confirming the association between the sequence of zeros and the natural numbers.
Is the Riemann conjecture considered correct by the author?
The author suggests that the Riemann conjecture may be undecidable or even incorrect, noting that it has been a target of numerous failed validation attempts by the mathematical community.
- Quote paper
- William Fidler (Author), 2022, The irrelevance of the location of Riemann's zeros to the disposition of the prime numbers, Munich, GRIN Verlag, https://www.grin.com/document/1191150
