Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...
who made contributions to analysis, number theory, and
differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential geometry of curves, theor ...
. In the field of
real analysis Image:Fourier Series.svg, 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. In mathematics, real analysis is the branch of mathematical analysis that studies the behavi ...
, he is mostly known for the first rigorous formulation of the integral, the
Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the
prime-counting function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, containing the original statement of the
Riemann hypothesis In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its Root of a function, zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved pro ...
, is regarded as one of the most influential papers in
analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (complex analysis), argument. ...
. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of
general relativity General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, ...
. He is considered by many to be one of the greatest mathematicians of all time.


Early years

Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the
Kingdom of Hanover The Kingdom of Hanover (german: Königreich Hannover) was established in October 1814 by the Congress of Vienna, with the restoration of George III to his Hanoverian territories after the Napoleonic era. It succeeded the former Electorate of Han ...
. His father, Friedrich Bernhard Riemann, was a poor
Lutheran Lutheranism is one of the largest branches of Protestantism that identifies with the teachings of Martin Luther, a 16th-century German Protestant Reformers, reformer whose efforts to reform the theology and practice of the church launched the Pro ...
pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.


During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years). After the death of his grandmother in 1842, he attended high school at the :de:Johanneum Lüneburg, Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, where he planned to study towards a degree in Theology. However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on the method of least squares). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.


Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Albert Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following the death of Dirichlet (who held Gauss's chair at the University of Göttingen), he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using higher dimensions, dimensions higher than merely three or four in order to describe physical reality. In 1862 he married Elise Koch and they had a daughter Ida Schilling who was born on 22 December 1862.

Protestant family and death in Italy

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania).
Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting the Lord’s Prayer with his wife and died before they finished saying the prayer. Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost forever. Riemann's tombstone in Biganzolo (Italy) refers to :

Riemannian geometry

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology and is still being applied in novel ways to mathematical physics. In 1853, Carl Friedrich Gauss, Gauss asked Riemann, his student, to prepare a ''Habilitationsschrift'' on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen in 1854 entitled