It is demonstrated in this work that we may construct an infinite number of strips in the complex plane having the same 'dimensions as the Critical Strip and which are devoid of Riemann zeros except on the line of symmetry. It is shown that the number of zeros on each line is infinite, indeed, there is a Riemann zero at infinity.
It is posited that a form of the Riemann conjecture is verified in each strip. It is shown that each integer in the infinite set of the integers has an associated Riemann zero and that the imaginary parts of the complex number at which the zeros are located are proportional to the 'local' asymptote to the prime counting function. A connection between the prime counting function and the zeta function is established. A limited distribution of the Riemann zeros corresponding to their respective prime numbers is constructed and it is seen that, at least over this range, the two are correlated, albeit non-linearly.
It is demonstrated that the imaginary part of the complex number locating a Riemann zero may, for any integer that can be articulated, be obtained by a few keystrokes of a hand calculator.
Table of Contents
Introduction
The analysis
The prime counting function
The Riemann Hypothesis
A Critical Strip at the trivial zeros
The connection of the Riemann zeros to the prime numbers
Discussion
Objectives & Research Themes
This work aims to challenge the traditional focus on the Critical Strip regarding the Riemann Hypothesis by proposing an analytical verification in the negative half-plane at the positions of the trivial zeros. The author seeks to establish a direct relationship between Riemann zeros, the integers, and the prime counting function.
- Critique of the "Critical Strip" as a mathematical graveyard.
- Analytical construction of Riemann zeros in the negative half-plane.
- Mathematical connection between the Riemann zeta function and prime numbers.
- Practical calculation of zero locations using simple arithmetic rather than supercomputers.
- Reinterpretation of the significance of trivial zeros.
Excerpt from the book
The Riemann Hypothesis
We have previously described the Critical Strip as a mathematical graveyard and for this reason we do not intend to repeat what we consider to be the misguided attempts of others in the examination of this region.
Rather, we consider that we have established a proof of Riemann’s Hypothesis, not in this region but in the negative half plane at the positions of the trivial zeros.
It is posited that the limited investigation of the trivial zeros by others may be due to the fact that they are easy to find and also that there may be a psychological aversion to them existing in the negative half-plane. They are generally dismissed as being without import.
From here, for brevity, we will refer to zero values of Riemann’s zeta function as Riemann zeros.
From what has been shown before it follows that at each trivial zero there is a line, extending ‘upwards’ to infinity along which an infinite set of Riemann zeros, corresponding to each of the integers, is located.
Chapter Summary
Introduction: The author introduces the Riemann Hypothesis as a pivotal unsolved problem and critiques the long-standing focus on the Critical Strip as unproductive.
The analysis: This section details the mathematical framework, employing the Riemann zeta function and Dirichlet eta function to explore zero locations.
The prime counting function: The chapter explores the asymptotic distribution of prime numbers and establishes a relationship between the imaginary part of s and the prime-counting function.
The Riemann Hypothesis: The author presents the core argument that the hypothesis can be verified in the negative half-plane through trivial zeros.
A Critical Strip at the trivial zeros: This chapter constructs a region in the negative half-plane mirroring the dimensions of the traditional Critical Strip.
The connection of the Riemann zeros to the prime numbers: The text establishes a formal connection between the zeta function and prime numbers via a modified counting function.
Discussion: The author summarizes the findings, reiterating that the Critical Strip is likely the wrong region for analytical verification.
Keywords
Riemann Hypothesis, Riemann zeta function, Critical Strip, Prime numbers, Prime counting function, Trivial zeros, Analytical continuation, Dirichlet eta function, Number theory, Complex plane, Mathematical proof, Imaginary part, Asymptotic distribution, Integers
Frequently Asked Questions
What is the primary subject of this work?
The work focuses on providing an alternative analytical approach to the Riemann Hypothesis, suggesting that it should be examined outside the traditional Critical Strip.
What are the central themes explored?
The core themes include the location of Riemann zeros, the relationship between these zeros and prime numbers, and the critique of modern computational approaches to the conjecture.
What is the ultimate goal of the author?
The primary goal is to demonstrate that the Riemann Hypothesis can be verified analytically at the positions of trivial zeros in the negative half-plane.
Which mathematical methods are primarily employed?
The author uses analytic continuation, the functional equation of the Riemann zeta function, and comparisons to the prime counting function to derive new relationships.
What does the main body of the work cover?
It covers the mathematical derivation of zero locations, the construction of a symmetric region in the negative half-plane, and the correlation between Riemann zeros and prime number density.
Which keywords best characterize this document?
Key terms include Riemann Hypothesis, Critical Strip, Prime counting function, Riemann zeros, and Analytical proof.
Why does the author refer to the Critical Strip as a "mathematical graveyard"?
The author uses this metaphor to describe the region where generations of mathematicians have unsuccessfully attempted to prove the hypothesis using brute-force computational methods.
How does the author calculate the position of Riemann zeros compared to standard methods?
While standard methods require massive supercomputing power to locate zeros in the Critical Strip, the author demonstrates that their proposed locations can be determined using a simple hand calculator.
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- William Fidler (Autor), 2021, On the Riemann Hypothesis, Múnich, GRIN Verlag, https://www.grin.com/document/993094
