Given their importance as the building blocks of all of the integers, the prime numbers, their number and dispositionwithin the integers have been studied for millennia. In 1859 Bernhard Riemann published a short paper (which only extended to six manuscript pages) concerned with an investigation of the number of prime numbers less than any given number. The outcome of the work revolutionised number theory and in the years following has resulted in almost what could be termed an industry in a particular aspect of his work, now called the Riemann Hypothesis. Riemann showed that all the zero values of the zeta function in the positive part of the complex plane should lie in a region between x = 0 and x = 1, and, in particular, conjectured that they all should lie on the line of symmetry x = ½.
This conjecture, which is the basis of many other conjectures in Number Theory is considered by many mathematicians to be the currently greatest unsolved problem in Mathematics—so much so that the Clay Mathematical Institute of Boston, Mass. has offered one million dollars to anyone who can produce a solution. No attempt is made here to verify the hypothesis, for its validity is not required. A procedure is developed here, by means of which, the zeros of Riemann's zeta function in the so-called Critical Strip of the complex plane may be assigned values of the prime number counting function. The procedure is novel and uses the concept of a Dirichlet line in the complex plane and a quantity called a nearodd (both of which are defined in the text).
The process may be rendered self-contained, in the sense that, when it is associated with Gram's series the only input required is the magnitude of the imaginary part of the function s = x + iy which will locate a Riemann zero on the Critical Line; vast numbers of zeros may be accessed in [2]. Further, it is shown that the Riemann conjecture is irrelevant in the assigning of any particular value of the prime number counting function to the corresponding Riemann zero. It is suggested, pace Wiles, who obtained a proof of Fermat's Last Theorem as a by-product of his verification of the Taniyama-Shimura conjecture, that, in the light of Godel's incompleteness theorems, Riemann's hypothesis may be undecidable.
Table of Contents
1. Introduction
2. Analysis
3. Dirichlet lines in the complex plane
4. The assigning of values of the prime number counting function to the Riemann zeros along the line passing through s = ½
5. The Riemann hypothesis
6. The Critical Strip
7. Discussion
Research Objectives and Key Topics
This work develops a novel procedure to assign values of the prime number counting function to the zeros of the Riemann zeta function within the critical strip of the complex plane, using the concept of Dirichlet lines and a quantity defined as a nearodd.
- Application of Dirichlet lines to locate Riemann zeros in the complex plane.
- Introduction and methodology of the "nearodd" quantity for mapping zeros.
- Evaluation of the relationship between Riemann's zeta function and the prime number counting function.
- Theoretical implications regarding the potential undecidability of the Riemann hypothesis.
- Analysis of the consistency of the prime number counting function across the complex plane.
Excerpt from the Book
Dirichlet lines in the complex plane
It follows from the preceding that, in the complex plane, along lines with an ordinate given by the formula: y = kπ/ln k, for any given constant k , where, of course k is an integer, the imaginary part of the zeta function is zero at all points. Along such a line the real part of the zeta function at any point is shown to be equal to the negative of the Dirichlet function for a real number.
We name these lines Dirichlet lines in honour of Dirichlet, for they bear, in effect, the distribution of the Dirichlet function η(x) in the range, -∞ ≤ η(x) ≤ +∞. Further, we see that the Riemann zeta function is zero whenever a Dirichlet line intersects a ‘critical line’ through a trivial zero of the zeta function. These remarks apply to all k.
Summary of Chapters
Introduction: Provides a historical overview of prime numbers and Riemann's landmark 1859 paper regarding the distribution of primes and the zeta function.
Analysis: Details the mathematical derivation, including the Riemann zeta function, Dirichlet eta function, and the behavior of these functions in the complex plane.
Dirichlet lines in the complex plane: Defines Dirichlet lines as specific ordinate-based lines where the imaginary part of the zeta function vanishes.
The assigning of values of the prime number counting function to the Riemann zeros along the line passing through s = ½: Explains the iterative process of linking the imaginary parts of Riemann zeros to prime counting function values using the nearodd parameter.
The Riemann hypothesis: Discusses the nature of the Riemann hypothesis in light of Gödel's incompleteness theorems and its potential status as undecidable.
The Critical Strip: Describes the division of the critical strip into infinite cells bounded by Dirichlet lines.
Discussion: Synthesizes the findings, confirming the utility of Dirichlet lines and the consistent nature of the prime number counting function throughout the complex plane.
Keywords
Riemann zeta function, Dirichlet lines, Critical Strip, Prime number counting function, Nearodd, Riemann hypothesis, Complex plane, Gram's series, Number theory, Trivial zeros, Non-trivial zeros, Prime numbers, Mathematical logic, Euler product formula.
Frequently Asked Questions
What is the fundamental subject of this research?
The work explores a mathematical procedure to associate specific values of the prime number counting function with the zeros of the Riemann zeta function.
What are the primary thematic areas covered?
The core themes include the distribution of Riemann zeros, the properties of Dirichlet lines, the application of the nearodd quantity, and the evaluation of prime counting functions.
What is the primary goal of this publication?
The goal is to demonstrate a novel, self-contained method for mapping Riemann zeros to prime counting values, independent of traditional prime distribution conjectures.
Which scientific methodology is employed?
The author employs analytical derivation, functional equations of the zeta and eta functions, and iterative numerical methods to relate complex coordinates to real-number indices.
What topics are discussed in the main body?
The main body focuses on the definition of Dirichlet lines, the iterative calculation of nearodds, the connection between these nearodds and prime counts, and the theoretical implications for the Riemann hypothesis.
How would you characterize this work through keywords?
It is best defined by Riemann zeta function, Dirichlet lines, nearodd, prime number counting function, and the critical strip.
How does the author suggest the Riemann hypothesis might be viewed in the context of logic?
The author suggests that in light of Gödel's incompleteness theorems, the Riemann hypothesis might be considered undecidable within formal axiomatic systems.
What role does the 'nearodd' play in the proposed method?
The nearodd acts as a crucial integer-based parameter derived from the imaginary part of the Riemann zero, allowing for the association with prime number counts.
Why is the first cell in the critical strip bounded by 3 and 5?
The author excludes 2 as a boundary because it is the only even prime, making it unsuitable for the definition of the nearodd within the matrix of odd numbers.
- Citar trabajo
- William Fidler (Autor), 2022, Assigning of values of the prime number counting function to Bernhard Riemann's zeros. Concept of Dirichlet lines in the complex plane, Múnich, GRIN Verlag, https://www.grin.com/document/1168625
