Bernhard Riemann

born in Breselenz, Germany 17 September 1826

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 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 knew of Pafnuty Chebyshev's work on the Prime Number Theorem. He had visited Dirichlet in 1852.

In 1853, Gauss asked his student Riemann 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 "Ueber die Hypothesen welche der Geometrie zu Grunde liegen" ("On the hypotheses which underlie geometry"). It was only published twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow but it is now recognized as one of the most important works in geometry.

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 Dirichlet's death, he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality. In 1862 he married Elise Koch and had a daughter.

He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behavior of closed paths about singularities (described by the monodromy matrix). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems.

According to Detlef Laugwitz, automorphic functions appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces.

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.

In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler), behind which a theta function lies. Also, it gives a better approximation for the prime-counting function π ( x ) {\displaystyle \pi (x)} than Gauss's logarithmic integral function Li ⁡ ( x ) {\displaystyle \operatorname {Li} (x)} . Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for π ( x ) {\displaystyle \pi (x)} .

In his habilitation work on Fourier series, where he followed the work of his Teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an Example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large n.