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.
Inhaltsverzeichnis (Table of Contents)
- Abstract
- Introduction
- Analysis
- Discussion
- References
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The main objective of this work is to extend previous research on the Riemann zeta function, focusing on the trivial zeros in the negative half of the complex plane. It introduces the concept of Dirichlet and Riemann lines and investigates their relationship to the zeros of the zeta function.
- The definition and properties of Dirichlet and Riemann lines in the complex plane.
- The location of zeros of the Riemann zeta function at the intersection of these lines and ordinate lines passing through trivial zeros.
- The relationship between the Riemann zeta function and Dirichlet's alternating function along these lines.
- An examination of the Riemann zeta function at the trivial zeros.
- The significance of even and odd multipliers associated with Riemann and Dirichlet lines, respectively.
Zusammenfassung der Kapitel (Chapter Summaries)
Abstract: This chapter introduces the concepts of Dirichlet and Riemann lines in the complex plane. It explains that these lines extend throughout the complex plane, and along them, the zeta function is defined by the negative of Dirichlet's alternating function (for Dirichlet lines) and the zeta function for a real number (for Riemann lines). The chapter highlights the infinite number of these lines and their connection to the zeros of the Riemann zeta function, particularly at the intersections with ordinate lines passing through trivial zeros. A key distinction is established between the lines based on their associated multipliers: odd for Dirichlet lines and even for Riemann lines.
Introduction: This chapter provides background on Riemann's 1859 paper and its relevance to predicting the number of prime numbers. It discusses Riemann's discovery of trivial zeros at -2q (where q = 1, 2, 3, etc.) and the non-trivial zeros within the critical strip. The Riemann Hypothesis, concerning the location of non-trivial zeros on the critical line, is introduced, along with its significance and the lack of analytical verification to date. The chapter emphasizes the relative lack of research on the trivial zeros and positions this work as contributing to a more balanced understanding of the zeta function.
Analysis: This chapter delves into the mathematical analysis of the Riemann zeta function, η(s), and their relationship. It begins by reiterating portions of previous work by the author. The chapter presents equations (1) and (2), defining ζ(s) and η(s), and derives equation (3), highlighting the connection between these functions. Riemann's functional equation is introduced, and the Dirichlet functional equation is expressed in terms of Riemann's functional equation. The chapter then focuses on finding a solution to equation (1) when ζ(s) = 0, illustrating the process by examining a term within the series expansion. Euler's theorem is used to simplify the expression, setting the stage for further analysis in subsequent sections (which are not included in the provided text).
Schlüsselwörter (Keywords)
Riemann zeta function, Dirichlet eta function, trivial zeros, non-trivial zeros, Riemann Hypothesis, Dirichlet lines, Riemann lines, complex plane, critical strip, critical line, analytical continuation, prime numbers.
Frequently Asked Questions about: A Comprehensive Language Preview of Research on the Riemann Zeta Function
What is the main topic of this research preview?
This preview summarizes research focusing on the Riemann zeta function, specifically its trivial zeros in the negative half of the complex plane. It introduces the novel concepts of Dirichlet and Riemann lines and investigates their relationship to the zeros of the zeta function.
What are the key objectives of the research?
The main objective is to extend previous research by investigating the behavior of the Riemann zeta function at its trivial zeros. This involves exploring the properties of Dirichlet and Riemann lines in the complex plane, their relationship to the zeros, and the connection between the Riemann zeta function and Dirichlet's alternating function along these lines. The significance of even and odd multipliers associated with Riemann and Dirichlet lines is also examined.
What are Dirichlet and Riemann lines?
Dirichlet and Riemann lines are lines extending throughout the complex plane. Along these lines, the zeta function is defined by the negative of Dirichlet's alternating function (for Dirichlet lines) and the zeta function for a real number (for Riemann lines). They are distinguished by their associated multipliers: odd for Dirichlet lines and even for Riemann lines. Their intersections with ordinate lines passing through trivial zeros are a key focus of the research.
What is the significance of the trivial zeros of the Riemann zeta function?
While much research focuses on the non-trivial zeros and the Riemann Hypothesis, this research highlights the relative lack of study on the trivial zeros located at -2q (where q = 1, 2, 3, etc.). The work aims to contribute to a more balanced understanding of the zeta function by investigating these often-overlooked zeros.
How does this research relate to the Riemann Hypothesis?
The preview provides background on the Riemann Hypothesis, which concerns the location of non-trivial zeros on the critical line. While not directly addressing the Riemann Hypothesis, this research contributes to a broader understanding of the Riemann zeta function, potentially offering insights that could indirectly inform future work on the Hypothesis.
What is the methodology used in this research?
The research involves a mathematical analysis of the Riemann zeta function (ζ(s)) and the Dirichlet eta function (η(s)), exploring their relationship and applying equations to understand the behavior of ζ(s) at its trivial zeros. The analysis includes using Euler's theorem to simplify expressions and examining terms within the series expansion of the zeta function.
What are the key findings (as presented in the preview)?
The preview highlights the definition and properties of Dirichlet and Riemann lines, their connection to the trivial zeros of the Riemann zeta function, and the relationship between the Riemann zeta function and Dirichlet's alternating function along these lines. It emphasizes the significant role of even and odd multipliers in distinguishing between Riemann and Dirichlet lines, respectively.
What are the key chapters covered in the preview?
The preview includes summaries of an Abstract, Introduction, Analysis chapter, along with a Table of Contents and a list of Keywords. The Analysis chapter details the mathematical analysis of the Riemann zeta function and its relationship with Dirichlet's eta function.
What are the key words associated with this research?
Key words include: Riemann zeta function, Dirichlet eta function, trivial zeros, non-trivial zeros, Riemann Hypothesis, Dirichlet lines, Riemann lines, complex plane, critical strip, critical line, analytical continuation, prime numbers.
- Quote paper
- William Fidler (Author), 2022, On zeros of Riemann's zeta function in the negative half of the complex plane. Dirichlet lines and the concept of Riemann lines, Munich, GRIN Verlag, https://www.grin.com/document/1240198
