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.
Inhaltsverzeichnis (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
- References
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This paper aims to demonstrate the direct connection between Riemann zeros, counting numbers, and prime numbers, challenging the established view of the Critical Strip's significance in this relationship. It argues that the Critical Strip is not the sole determinant of prime number distribution.
- The relationship between Riemann zeros, counting numbers, and prime numbers
- The irrelevance of the Critical Strip for understanding prime number distribution
- The importance of 'critical lines' other than the Critical Strip
- The limitations of Riemann's conjecture and the Critical Strip
- The intrinsic nature of prime number distribution within the system of counting numbers
Zusammenfassung der Kapitel (Chapter Summaries)
The introduction sets the stage by outlining Riemann's 1859 work and its focus on the Critical Strip. It highlights the ongoing efforts to verify Riemann's conjecture, emphasizing the concentration on the Critical Strip and the dismissal of other regions in the complex plane. The paper then re-introduces key concepts from previous work by the author, including the relationship between the Riemann zeta function, Dirichlet eta function, and their functional equations. The analysis delves into the nature of Riemann zeros and their relationship to counting numbers, arguing that the Critical Strip is not the sole determinant of the prime number counting function. Instead, the paper proposes that the location of Riemann zeros on 'critical lines' other than the Critical Strip is crucial for understanding the distribution of prime numbers. This chapter also underscores the intrinsic nature of prime number distribution within the system of counting numbers, suggesting that the distribution of Riemann zeros is not a causal factor.
Schlüsselwörter (Keywords)
Riemann zeros, prime numbers, Critical Strip, counting numbers, prime number counting function, Dirichlet eta function, Riemann zeta function, critical lines, mathematical graveyard, analytic continuation.
- 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.hausarbeiten.de/document/1191150
