In 1859, a German mathematician who worked at the University of Göttingen, Bernhard Riemann, published a short 8-page paper titled "Über die Anzahl der Primzahlen unter einer gegebenen Größe" ("On the Number of Prime Numbers Less Than a Given Quantity").
In this concise paper, Riemann introduced the idea of extending the zeta function ζ(s) into the complex plane and conjectured about the distribution of its zeros, which fundamentally relates to the distribution of prime numbers.
In the middle of the paper, when he was discussing the roots of one version of the zeta function, he wrote:
Original German
"……Man findet nun in der Tat etwa so viel reelle Wurzeln innerhalb dieser Grenzen, und es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien."
"……One indeed finds approximately as many real roots within these bounds, and it is very probable that all roots are real. However, a rigorous proof would be desirable; I have, nevertheless, left the search for it aside after some fleeting unsuccessful attempts, since it seemed dispensable for the next purpose of my investigation."
These sentences presented an intuitive yet unproven insight by Riemann on the properties of the roots of the zeta function. They were not presented as a formal conjecture, but have captured many mathematicians' attention.
Nearly 41 years later, on 8th August, 1900, David Hilbert, another German mathematician who also worked at Göttingen, stood up before the 200-odd delegates at the International Congress of Mathematicians in Paris. He presented his list of 23 "Mathematical Problems," and number 8 was this conjecture. This, without a doubt, contributed significantly to its prominence in the world mathematical community.
Another 100 years had elapsed, and as humanity stepped into a new millennium, the world had seen significant transformations from nearly every aspect. Yet, amidst these changes, Riemann's conjecture remains unproven.
In the year 2000, the Clay Mathematics Institute unveiled the Millennium Prize Problems, a collection of seven unsolved problems that were selected to highlight areas of active and important research in the field. One million dollars were offered for each problem solved. Among these problems is Riemann's conjecture.
Since then, the sentences mentioned in passing as part of Riemann's exploration of the zeta function's properties has been arguably the most famous of all unsolved mathematical problems, or "the Holy Grail of mathematics," and proving it has become the hardest way to earn 1 million dollars.
Mathematicians have summarized Riemann's sentences into one clear sentence and named it:
I stumbled upon the Riemann Hypothesis quite accidentally, and I only began delving into it at the same age Riemann was when he wrote the paper. Despite consistently excelling in exams, I found mathematics boring in school, thus I had not engaged with the subject since graduating from high school. But life always has a way of gifting you unexpected pathways. Somewhat fortuitously, I started tutoring high school students in maths during my free time. This not only refreshed my knowledge of high school maths, but also ignited a newfound passion for the subject. I began to see the inherent beauty in mathematics, which led me to embark on a journey of self-teaching.
During a lesson, one of my curious students asked me, "What is the Riemann Hypothesis?" I had to be honest and admit that although I had heard of it, I knew nothing about it. To adequately respond to the student's question, I researched the Riemann Hypothesis but quickly realized that I couldn't grasp much of it due to my lack of relevant skills and knowledge. That moment marked the beginning of my serious pursuit of learning Mathematical Analysis.
"If you had asked Euler, or even the young Gauss, what analysis was all about, he would have said: 'It is about the infinite and the infinitesimal.'"
Analysis is a branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions; it is studied usually in the context of real and complex numbers and functions.
Bernhard Riemann, in his relatively short life span of 40 years, had expanded the field significantly and profoundly influenced various areas, including complex analysis, differential geometry, and the theory of functions.
Riemann's approaches were usually innovative, combining mathematical rigor and intuitive leaps; his papers were known for density and conciseness. His writing style has been noted — or criticised — for its lack of clarity, making it difficult to follow. While his work is crucial and transformative, it assumes a rather high level of mathematical knowledge and does not explicitly detail each step of the proof, which may not align with modern expectations for clarity and accessibility.
Therefore, I decided to meticulously decipher his paper on the famous Riemann Hypothesis, analyzing it sentence by sentence to gain a detailed yet holistic understanding of the paper. I also intend to use this paper as a means to deepen my knowledge of analysis and number theory, and as a window to explore the historical development of mathematics.
I will be documenting my journey through a series of written pieces.
In each of my writings, I will concentrate on three areas:
Since I am writing while concurrently reading the paper, I will continuously revise my work whenever new insights arise or errors are detected.
If this was worth a few dollars to you, the jar is here. Either way, thank you for reading.
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