# Bernhard Riemann

**RiemannGeorg Friedrich Bernhard RiemannRiemann, BernhardRiemanRiemann's theoremsB. RiemannG. F. B. RiemannG. F. Bernhard RiemannG.F.B. RiemannGeorg Bernhard Riemann**

Georg Friedrich Bernhard Riemann ( 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.wikipedia

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### Riemann integral

**Riemann integrableRiemann-integrableintegrable**

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.

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

### On the Number of Primes Less Than a Given Magnitude

**1859 paper1859 paper on the zeta-functiona single short paper**

His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory.

"Ueber [sic] die Anzahl der Primzahlen unter einer gegebenen Grösse [sic]" (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.

### Fourier series

**Fourier coefficientFourier expansionFourier coefficients**

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.

Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision and formality.

### Mathematical analysis

**analysisclassical analysisanalytic**

Georg Friedrich Bernhard Riemann ( 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

In the middle of the 19th century Riemann introduced his theory of integration.

### Complex analysis

**complex variablecomplex functioncomplex functions**

His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.

Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century.

### Prime-counting function

**prime counting functioncounting prime numbersdenoting the number of prime numbers**

His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory.

The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859.

### Analytic number theory

**analyticanalytic number theoristanalytic techniques**

His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory.

Bernhard Riemann made some famous contributions to modern analytic number theory.

### Number theory

**number theoristcombinatorial number theorytheory of numbers**

Georg Friedrich Bernhard Riemann ( 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

The use of complex analysis in number theory comes later: the work of Bernhard Riemann (1859) on the zeta function is the canonical starting point; Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (modular forms).

### Riemannian geometry

**Riemannianlocal to global theoremsRiemann geometry**

Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture " Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based").

### General relativity

**general theory of relativitygeneral relativity theoryrelativity**

Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity.

The 19th century mathematician Bernhard Riemann's non-Euclidean geometry, called Riemannian Geometry, provided the key mathematical framework which Einstein fit his physical ideas of gravity on, and enabled him to develop general relativity.

### Differential geometry

**differentialdifferential geometerdifferential geometry and topology**

Georg Friedrich Bernhard Riemann ( 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way.

### Riemann surface

**Riemann surfacescompact Riemann surfaceconformally invariant**

His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.

These surfaces were first studied by and are named after Bernhard Riemann.

### Carl Friedrich Gauss

**GaussCarl GaussCarl Friedrich Gauß**

However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on the method of least squares).

In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (About the hypotheses that underlie Geometry).

### Riemann curvature tensor

**Riemann tensorcurvature tensorcurvature**

The fundamental object is called the Riemann curvature tensor.

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds.

### Dimension

**dimensionsdimensionalone-dimensional**

He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality.

Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann.

### Cauchy–Riemann equations

**Cauchy–Riemann operatorCauchy–Riemann equationCauchy–Riemann conditions**

Complex functions are harmonic functions (that is, they satisfy Laplace's equation and thus the Cauchy–Riemann equations) on these surfaces and are described by the location of their singularities and the topology of the surfaces.

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic.

### Topology

**topologicaltopologicallytopologist**

This area of mathematics is part of the foundation of topology and is still being applied in novel ways to mathematical physics.

Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.

### Peter Gustav Lejeune Dirichlet

**DirichletLejeune DirichletGustav Dirichlet**

During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching.

Dirichlet enjoyed his time in Göttingen, as the lighter teaching load allowed him more time for research and, also, he got in close contact with the new generation of researchers, especially Richard Dedekind and Bernhard Riemann.

### Riemannian manifold

**Riemannian metricRiemannianRiemannian manifolds**

This is the famous construction central to his geometry, known now as a Riemannian metric.

These terms are named after the German mathematician Bernhard Riemann.

### Carl Gustav Jacob Jacobi

**JacobiCarl Gustav JacobiCarl Jacobi**

During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching.

This method of inversion, and its subsequent extension by Weierstrass and Riemann to arbitrary algebraic curves, may be seen as a higher genus generalization of the relation between elliptic integrals and the Jacobi or Weierstrass elliptic functions.

### Tensor

**tensorsorderclassical treatment of tensors**

Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe how much it was bent or curved.

Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus.

### Non-Euclidean geometry

**non-Euclideannon-Euclidean geometriesalternative geometries**

For the surface case, this can be reduced to a number (scalar), positive, negative, or zero; the non-zero and constant cases being models of the known non-Euclidean geometries.

Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature.

### Gotthold Eisenstein

**EisensteinEisenstein, GottholdFerdinand Eisenstein**

During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching.

Bernhard Riemann attended his classes on elliptic functions.

### Jameln

**Breselenz**

Riemann was born on September 17, 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover.

Mathematician Bernhard Riemann (1826–1866) was born in Breselenz.

### Dannenberg (Elbe)

**DannenbergBrunswick-DannenbergDannenberg Ost**

Riemann was born on September 17, 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover.