Riemann was educated at Gottingen University in Brunswick, Germany.
The presence of the legendary Gauss made this university the centre of the
mathematical world. Riemann was one of the few great mathematicians who was also
a pleasant person! He was shy and, when speaking in public, nervous.At age thirty-three, after many years of poverty, he became a full professor at Gottingen. In poor health, he died at age thirty-nine.
Riemann made significant contributions to the theory of functions of a complex variable and developed the concept of Riemann sums as the basis for the Integral.
Before being appointed an unpaid lecturer, candidates at Gottingen were required to name three topics in which they had some competence and lecture on the first such topic. Riemann listed his third topic as the Foundations of Geometry. Unfortunately, this was a topic of great interest to Gauss who insisted that Riemann give his lecture on this topic. Unprepared, Riemann devoted himself to geometry for two months and gave what is often regarded as the most important scientific lecture ever given. The field of Riemannian Geometry was born. Even Gauss (who rarely praised his contemporaries) was pleased. Later, Riemann's analysis was used as the basis for Einstein's Theory of General Relativity.
In later years, Riemann made significant contributions to number theory.
In particular, this research led to the Riemann zeta Function:
z(z) = 1 + 1/2z + 1/3z + 1/4z + ...
(where z = x + iy is a complex number) and one of the most important unsolved
problems in modern mathematics:
"In the interval 0 < x < 1, the only zeros of
z(z) occur when x = 1/2.