Georg Friedrich Bernhard Riemann (German: [ˈʀiːman] ( listen); 17 September 1826 – 20 July 1866) was an influential German mathematician who made lasting and revolutionary contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, 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, containing the original statement of the Riemann hypothesis, is regarded, although it is his only paper in the field, as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity.
Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today. His father, Friedrich Bernhard Riemann, was a poor Lutheran 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). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. His teachers were amazed by his adept 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 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 renowned 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 (but he remained a devout Christian throughout his life); after getting his father's approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Jacobi, Lejeune Dirichlet, Steiner, and 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 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 Lejeune Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality—an idea that was ultimately vindicated with Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.
Austro-Prussian War 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 Romans 8:28 ("And we know that all things work together for good to them that love God, to them who are the called according to his purpose"):
Here rests in God Georg Friedrich Bernhard Riemann
Professor in Göttingen
born in Breselenz, 17 September 1826
died in Selasca, 20 July 1866
Those, who love God, all things must serve to its best manner.
Four- / other-dimensional