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Riemann did a lot of work in the theory of complex functions in his PhD thesis. So he derived the Cauchy-Riemann equations ([Werke, p.6]) and the Riemann mapping theorem and developed the idea of a Riemann surface and multiple planes. The Riemann Hypothesis, which he statet in ``Über die Anzahl der Primzahlen unter einer gegebenen Grösse", Monatsberichte der Berliner Akademie, November 1859, about the zero roots of the is not solved yet. In his Habilitationsvortrag: ``Uber die Hypothesen, welche der Geometrie zugrundeliegen" (first published in 1867 in Göttingen), he developed the notion of an n-dimensional manifold and tried to set a metric upon it. In this context he constructed the Riemannian normal coordinates.
He also constructed the Riemann integral and found a necessary and a sufficient condition for a function to be integrable. [Katz]