An infinite set of hybrid functions with one unique member whose verifiable zeros are to be found only on Riemann's Critical Line and nowhere else in the Critical Strip

This paper devises a hybrid function, denoted by H_a, (where a, is a real constant), which consists of the linear combination of a novel form of the Riemann zeta function and the abscissa of any point in the complex plane. These functions comprise an infinite set, for the value and algebraic sign of the constant are unconstrained.

Amongst these functions, H_(1⁄2) is unique, in that, the magnitude of its value at the intersection of any Dirichlet line [1] with Riemann’s Critical Line [2] is shown to be absolutely zero and that there are no other zeros of this function anywhere else in the Critical Strip.

There may be other zeros of this function elsewhere in the complex plane, but this paper argues that this can never be proved; this is a feature of any other of the H_a whose zeros can be posited to exist at the intersection of a vertical line passing through any abscissa of choice with a Dirichlet line but, can never be shown to be exactly zero, since this would require that the Dirichlet alternating eta series associated with the real part of these H_a be summed to infinity.

It follows from the above that, for the function, H_(1⁄2) Riemann’s hypothesis is verified.

Inhaltsverzeichnis (Table of Contents)

• Abstract
• Introduction
• Analysis
  • The function Fx + i yk
  • The hybrid function
  • Riemann's zeros and the hybrid function H1/2
  • The trivial zeros and the hybrid function
  • The Critical Points
• Discussion
• References

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

The objective of this work is to introduce a hybrid function denoted by Ha, which consists of a linear combination of a novel form of the Riemann zeta function and the abscissa of any point in the complex plane. The author explores the unique properties of this function, particularly H1/2, and its implications for Riemann's Hypothesis.

• The properties of the hybrid function Ha
• The uniqueness of the function H1/2
• The relationship between the hybrid function and Riemann's Hypothesis
• The distribution of zeros of the hybrid function
• The use of numerical calculations to explore the function's behavior

Zusammenfassung der Kapitel (Chapter Summaries)

• Introduction: This chapter provides a historical overview of the Riemann zeta function and its significance in number theory. It highlights the importance of Riemann's Hypothesis and the search for functions that might shed light on its validity.
• Analysis: This chapter delves into the mathematical analysis of the hybrid function Ha. It defines the function, explores its connection to the Riemann zeta function and Dirichlet eta function, and examines the distribution of its zeros. The chapter also introduces the concept of Dirichlet lines and their intersection with the Critical Line.

Schlüsselwörter (Keywords)

The key terms and concepts of this work include: hybrid function, Riemann zeta function, Dirichlet eta function, Riemann's Hypothesis, Critical Line, Dirichlet lines, zeros, complex plane, numerical calculations, analytical verification.

Details

• Grade: 2.0
• Author: William Fidler (Author)
• Publication Year: 2022
• Pages: 18
• Catalog Number: V1272674
• ISBN (eBook): 9783346725103
• ISBN (Book): 9783346725110
• Language: English