# Riemann, Georg Friedrich Bernhard

### theory mathematics physics riemann’s

[ree man] (1826–66) German mathematician: originated Riemannian geometry.

Riemann, the son of a Lutheran pastor, studied theology to please his father, and then studied mathematics under at Göttingen to please himself. In 1859 he became professor of mathematics there. At the age of 39 he died of tuberculosis. His friend said of Riemann ‘The gentle mind which had been implanted in him in his father’s house remained with him all his life, and he served his God faithfully, as his father had, but in a different way.’

Riemann’s papers were few but perfect, even in Gauss’s eyes, producing profound consequences and new areas of mathematics and physics. Riemann’s earliest publication was a new approach to the theory of complex functions using potential theory (from theoretical physics) and geometry to develop Riemann surfaces, which represent the branching behaviour of a complex algebraic function. These ideas were extended by introducing topological concepts into the theory of functions; this work was developed by to advance algebraic geometry. In another paper Riemann defined a function *f* (s), the Riemann zeta function, where

where *s* = *u* + *i* v is complex, and conjectured that *f* ( *s* )= 0 only if *u* =½ for 0 < *u* <1. No-one has proved Riemann’s hypothesis and it remains one of the important unsolved problems in number theory and analysis. In 2000 the Clay Mathematics Institute of Cambridge, MA, announced a 1 million dollar prize for a proof of it. Another of Riemann’s contributions to analysis was the introduction of the Riemann integral, defined in terms of the limit of a summation of an infinity of ever smaller elements.

In 1854 Riemann gave his inaugural lecture, ‘Concerning the hypotheses which underlie geometry’: a mathematical classic. The content was so fruitful that it altered mathematics and physics for a century afterwards. Riemann considered how concepts like distance and curvature could be defined generally in *n* -dimensional space, extending Gauss’s work (1827) on non-Euclidian geometries. He foresaw how important this was for physics and provided some of the mathematical tools for to construct his general theory of relativity (1915).

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