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.
Table of Contents
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
Objectives & Research Themes
This work aims to evaluate a newly devised hybrid function, H_a, and to investigate its zeros in the complex plane, ultimately providing a verification of the Riemann hypothesis for the unique member H_1/2.
- Investigation of a hybrid function formed by the linear combination of the Riemann zeta function and the abscissa.
- Proof that the zeros of H_1/2 lie exclusively on Riemann’s Critical Line within the Critical Strip.
- Analysis of the relationship between Dirichlet lines and the zeros of the hybrid function.
- Use of numerical calculations to explore the behavior of various hybrid functions as probabilities.
- Discussion of the uniqueness of H_1/2 in terms of absolute precision of its zeros.
Excerpt from the Book
The hybrid function
We now set down, by fiat, a hybrid function, H_a given by H_a = G[(a - x) + i y_k] + x (10). This expression may be taken to represent an infinite set of equations if we place no restriction on the value or algebraic sign of the constant, a. We stipulate that G is a zeta function, and so we may then rewrite this equation in the same form as that which produced equation (7); i.e. H_a = 1/(a-x)+iπ/ln 1 + 1/(a-x)+i2π/ln 2 + 1/(a-x)+i3π/ln 3 + ... 1/(j-1)(a-x)+(j-1)π/ln(j-1) + 1/(a-x)+iy_j + 1/(j+1)(a-x)+i(j+1)π/ln(j+1) + ............. + x (7).
Let us set y_j = jπ/ln j . As we have shown before, the function G will now become real, the imaginary terms having vanished and, the real part given by the negative of Dirichlet’s alternating eta function; hence, H_a becomes; H_a = -η(a - x) + x (11). Setting a = ½ we see that at x = ½, H_1/2 becomes, -η(0) + 1/2. Now, η(0) is Grandi’s series, and it was shown in [4] that this series has the value of 1/2, if, and only if, the number of terms in the series is infinite.
Summary of Chapters
Introduction: Outlines the historical context of the zeta function, Riemann’s work, and the motivation for investigating new hybrid functions whose zeros lie on the Critical Line.
Analysis: Provides a mathematical derivation of the Dirichlet eta function in relation to the Riemann zeta function and establishes the functional foundation for the work.
The function Fx + i yk: Describes the existence of Dirichlet lines within the complex plane and introduces the novel zeta function form.
The hybrid function: Introduces the definition of the hybrid function H_a and demonstrates how setting specific parameters leads to the verification of properties at x = 1/2.
Riemann’s zeros and the hybrid function H1/2.: Explores how the zeros of the hybrid function relate to Riemann zeros and identifies the uniqueness of H_1/2.
The trivial zeros and the hybrid function: Examines the behavior of trivial zeros and emphasizes the limitations of analytical verification for other hybrid functions.
The Critical Points: Defines Critical Points as the intersection of Dirichlet and Critical lines and discusses their association with counting numbers.
Discussion: Summarizes the findings, reflecting on the uniqueness of the hybrid function H_1/2 and the implications of the results.
Keywords
Riemann hypothesis, Riemann zeta function, Dirichlet etafunction, hybrid function, complex plane, Critical Line, Critical Strip, zeros, Dirichlet lines, absolute precision, mathematical proof, infinite series, Grandi's series, complex numbers, numerical calculation.
Frequently Asked Questions
What is the primary focus of this work?
The work focuses on a new set of hybrid functions, derived from the Riemann zeta function, to investigate the distribution of their zeros in the complex plane.
What are the central themes discussed?
The central themes include the relationship between Dirichlet lines and the Critical Line, the analytic properties of hybrid functions, and the verification of the Riemann hypothesis for a specific case.
What is the core objective or research question?
The core objective is to show that a specific hybrid function, H_1/2, has zeros that lie only on Riemann’s Critical Line, thereby verifying the Riemann hypothesis for this specific function.
What scientific methods are employed?
The author uses analytical derivation of functional equations, the application of series expansions, and numerical calculations to demonstrate the behavior of the hybrid functions.
What is covered in the main section of the book?
The main section covers the mathematical analysis of the zeta and eta functions, the definition of the hybrid function, potential zeros and their locations, and the discussion of Critical Points.
Which keywords best characterize this research?
The research is best characterized by terms such as Riemann hypothesis, hybrid function, Critical Line, Dirichlet etafunction, and complex plane analysis.
Why is the function H_1/2 considered unique?
H_1/2 is unique because its value at the intersection of a Dirichlet line and Riemann’s Critical Line can be determined with absolute precision, unlike other functions in the set.
How does the author interpret the role of 'failures' in the research?
The author ironically notes that the objectives were achieved by consistently examining the limitations and 'failures'—cases where values could not be determined with absolute certainty—thereby isolating the unique, solvable case.
- Quote paper
- William Fidler (Author), 2022, 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, Munich, GRIN Verlag, https://www.grin.com/document/1272674
