Bernhard riemann habilitation dissertation

Each of the intervals { J ( ε 1 ) i } has an empty intersection with X ε 1 , so each point in it has a neighborhood with oscillation smaller than ε 1 . These neighborhoods consist of an open cover of the interval, and since the interval is compact there is a finite subcover of them. This subcover is a finite collection of open intervals, which are subintervals of J ( ε 1 ) i (except for those that include an edge point, for which we only take their intersection with J ( ε 1 ) i ) . We take the edge points of the subintervals for all J ( ε 1 ) i s , including the edge points of the intervals themselves, as our partition.

In der Arbeit von Riemann sind noch viele weitere interessante Entwicklungen. So bewies er die Funktionalgleichung der Zetafunktion (die schon Euler bekannt ist), hinter der eine solche der Thetafunktion steckt. Auch gibt er eine viel bessere Näherung für die Primzahlverteilung π ( x ) {\displaystyle \pi (x)} als die Gauß’sche Funktion Li(x). Durch Summation dieser Näherungsfunktion über die nichttrivialen Nullstellen auf der Geraden mit Realteil 1/2 gibt er sogar eine exakte „explizite Formel“ für π ( x ) {\displaystyle \pi (x)} .

One of his teachers, Herr Schmalfuss, recognized Bernhard’s flair and began lending him advanced college-level mathematics texts, including works by Leonhard Euler and Adrien-Marie Legendre. The first time he did this, Herr Schmalfuss was astonished when Bernhard, after just a few days, returned the book to him. He questioned Bernhard about the book’s themes, and it became clear that his student truly had read and understood mathematical material that a typical advanced college student would have taken weeks or months to absorb.