Prologue
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 into the complex plane and conjectured about the distribution of its zeros, which fundamentally relates to the distribution of prime numbers.
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.
English Translation
……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.
The Riemann Hypothesis
Mathematicians have summarized Riemann’s sentences into one clear sentence:
“Alle Wurzeln [der Zeta-Funktion], die reellen Theil haben, gleich 1/2 sind.”
All non-trivial zeros of the zeta function have real part .
The Journey
I stumbled upon the Riemann Hypothesis quite accidentally. Despite consistently excelling in exams, I found mathematics boring in school. Somewhat fortuitously, I started tutoring high school students, which ignited a newfound passion for the subject.
During a lesson, a curious student asked: “What is the Riemann Hypothesis?” My inability to grasp it then marked the beginning of my serious pursuit of Mathematical Analysis.
Three Areas of Concentration
- The Language: My own German-to-English translations and stylistic analysis.
- The History: Context on Göttingen, David Hilbert, and the Millennium Prize.
- The Mathematics: Proofs and lemmas accessible to competent high school graduates.
Jetzt beginnen wir zusammen die spannende Reise!