The concept of a Dirichlet line in the complex plane was developed in [1]. This analysis is here extended to define another line in the complex plane, called by the author, a Riemann line. These lines are shown to extend throughout the whole of the complex plane. Along Dirichlet lines the zeta function is given by the negative of Dirichlet's alternating function for a real number, whilst along a Riemann line the zeta function is given by the zeta function for a real number. It is shown that there are an infinite number of these lines in the complex plane and, at the intersection of which with an ordinate line passing through any of the trivial zeros of the Riemann zeta function a zero of a Riemann zeta function is located.
A distinguishing characteristic of the Dirichlet lines and the Riemann lines is that they are associated with a multiplier which is an odd number for a Dirichlet line and an evev number for a Riemann line.
Table of Contents
1. Introduction
2. Analysis
3. The product (nk)
4. A hierarchy of infinities
5. The Riemann Hypothesis
6. Discussion
7. References
Objectives & Core Topics
The primary objective of this work is to provide a detailed examination of the Riemann zeta function at its trivial zeros and to introduce the concepts of Dirichlet lines and Riemann lines to describe the distribution of the function throughout the complex plane.
- Theoretical analysis of Riemann zeta and Dirichlet eta functions.
- Development of the concept of Dirichlet and Riemann lines as horizontal lines in the complex plane.
- Mathematical derivation of zeros on lines passing through trivial Riemann zeros.
- Investigation into the relationship between the distribution of prime numbers and Riemann zeros.
Excerpt from the Book
Analysis
Since the purpose of this work is to extend and elaborate on previous work [3] by the author it is considered essential to repeat part of the analysis shown therein.
The Riemann zeta function, 𝜁(s) is an extension to the series: 𝜁(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + 1/5^s + ---- (1). Here, the real number exponent is replaced with a complex number, s = x + i y. It should be noted that we use Riemann’s notation for the complex number but the normal mathematical notation for its real and imaginary parts.
Under the same constraint as above we write the Dirichlet eta function, 𝜂(s) as : 𝜂(s) = 1/1^s - 1/2^s + 1/3^s - 1/4^s + 1/5^s - --------------------------- (2). From equations (1) and (2) we get: 𝜁(s) - 𝜂(s) = 2^(1-s) [1/1^s + 1/2^s + 1/3^s + 1/4^s + ---- ] = 2^(1-s) 𝜁(s). For reasons which will become apparent later in the analysis we write the above as: -𝜂(s) = (2^(1-s) - 1) 𝜁(s) --------------------------------(3).
Summary of Chapters
Introduction: This chapter provides context on Riemann's original 1859 paper, the definition of trivial and non-trivial zeros, and the current status of the Riemann Hypothesis.
Analysis: This section conducts a mathematical evaluation of the zeta and eta functions, leading to the definition of Dirichlet and Riemann lines.
The product (nk): This chapter discusses the generation of zero locations using the product of variables n and k and demonstrates how the type of line (Dirichlet or Riemann) is identified by these parameters.
A hierarchy of infinities: This section asserts the existence of a hierarchy of infinities based on the different series of points identified for various n and k values.
The Riemann Hypothesis: This chapter contextualizes Riemann's conjecture within current mathematical efforts and disputes the link between prime number disposition and Riemann zeros.
Discussion: The final chapter summarizes the findings regarding the two types of horizontal lines and clarifies the utility of the derived zeta function in the study of prime numbers.
References: This section lists the academic sources and previous publications by the author that support the analysis.
Keywords
Riemann zeta function, Dirichlet lines, Riemann lines, trivial zeros, complex plane, Critical Strip, Critical Line, Prime numbers, Dirichlet eta function, Riemann Hypothesis, Coordinate geometry, Mathematical analysis, Functional equation, Series expansion.
Frequently Asked Questions
What is the core subject of this publication?
The work investigates the behavior of the Riemann zeta function, specifically focusing on its trivial zeros and introducing a new classification of lines ("Dirichlet lines" and "Riemann lines") within the complex plane.
What are the central thematic fields?
The book covers complex analysis, the properties of the Riemann zeta function, the relationship between prime number distribution and function zeros, and the geometry of lines in the complex plane.
What is the primary goal of the research?
The goal is to redress the imbalance in mathematical research that has heavily focused on the Critical Strip by providing a detailed examination of the zeta function at the locations of its trivial zeros.
Which scientific methodology is applied?
The author uses analytical continuation and iterative mathematical procedures to extend known functional structures and map the behavior of the zeta function across different ordinate lines.
What topics are covered in the main body?
The main body covers the formal definitions of zeta and eta functions, the derivation of lines using the parameters n and k, the visualization of these lines, and the refutation of the standard view linking prime number disposition to the Critical Strip.
Which keywords best characterize this work?
The most relevant keywords include Riemann zeta function, Dirichlet lines, Riemann lines, trivial zeros, and complex plane analysis.
How is a Dirichlet line distinguished from a Riemann line in this work?
Dirichlet lines are associated with an odd multiplier (n), while Riemann lines are associated with an even multiplier (n) as defined by the author's mathematical framework.
Does the author believe the Riemann Hypothesis is inherently tied to prime numbers?
No, the author argues that the disposition of prime numbers is an intrinsic characteristic of the counting system, and the association with Riemann zeros is incidental rather than causal.
- Arbeit zitieren
- William Fidler (Autor:in), 2022, On zeros of Riemann's zeta function in the negative half of the complex plane. Dirichlet lines and the concept of Riemann lines, München, GRIN Verlag, https://www.grin.com/document/1240198
