Bernhard Riemann
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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 in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
who made contributions to
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. In the field of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, he is mostly known for the first rigorous formulation of the integral, the
Riemann integral 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. It was presented to the faculty at the University of G ...
, and his work on Fourier series. His contributions to
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
include most notably the introduction of
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
, is regarded as a foundational paper of
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
. 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 and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. He is considered by many to be one of the greatest mathematicians of all time.


Biography


Early years

Riemann was born on 17 September 1826 in
Breselenz Jameln is a municipality in the district Lüchow-Dannenberg, in Lower Saxony, Germany. Jameln is part of the ''Samtgemeinde'' ("collective municipality") Elbtalaue. The main village in the municipality is Jameln, with around 450 inhabitants. Sett ...
, 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, identifying primarily with the theology of Martin Luther, the 16th-century German monk and Protestant Reformers, reformer whose efforts to reform the theology and practice of the Cathol ...
pastor in Breselenz who fought in the
Napoleonic Wars The Napoleonic Wars (1803–1815) were a series of major global conflicts pitting the French Empire and its allies, led by Napoleon I, against a fluctuating array of European states formed into various coalitions. It produced a period of Fren ...
. 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 talent, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.


Education

During 1840, Riemann went to
Hanover Hanover (; german: Hannover ; nds, Hannober) is the capital and largest city of the German state of Lower Saxony. Its 535,932 (2021) inhabitants make it the 13th-largest city in Germany as well as the fourth-largest city in Northern Germany ...
to live with his grandmother and attend
lyceum The lyceum is a category of educational institution defined within the education system of many countries, mainly in Europe. The definition varies among countries; usually it is a type of secondary school. Generally in that type of school the t ...
(middle school years). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the
Bible The Bible (from Koine Greek , , 'the books') is a collection of religious texts or scriptures that are held to be sacred in Christianity, Judaism, Samaritanism, and many other religions. The Bible is an anthologya compilation of texts of a ...
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 Philology () is the study of language in oral and writing, written historical sources; it is the intersection of textual criticism, literary criticism, history, and linguistics (with especially strong ties to etymology). Philology is also defin ...
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 The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
, where he planned to study towards a degree in
theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...
. However, once there, he began studying
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
under
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(specifically his lectures on the
method of least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
). 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 Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative ...
in 1847. During his time of study,
Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
,
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
,
Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
, and
Gotthold Eisenstein Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician. He specialized in number theory and analysis, and proved several results that eluded even Gauss. Like Galois and Abel before him, Eisenstein died ...
were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.


Academia

Riemann held his first lectures in 1854, which founded the field of
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
and thereby set the stage for
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's
general theory of relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
. 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 Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
'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 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 Hanover (; german: Hannover ; nds, Hannober) is the capital and largest city of the German state of Lower Saxony. Its 535,932 (2021) inhabitants make it the 13th-largest city in Germany as well as the fourth-largest city in Northern Germany ...
and
Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an em ...
clashed there in 1866. He died of
tuberculosis Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, in ...
during his third journey to Italy in Selasca (now a hamlet of
Verbania Verbania (, , ) is the most populous ''comune'' (municipality) and the capital city of the province of Verbano-Cusio-Ossola in the Piedmont region of northwest Italy. It is situated on the shore of Lake Maggiore, about north-west of Milan and ...
on
Lake Maggiore Lake Maggiore (, ; it, Lago Maggiore ; lmo, label=Western Lombard, Lagh Maggior; pms, Lagh Magior; literally 'Greater Lake') or Verbano (; la, Lacus Verbanus) is a large lake located on the south side of the Alps. It is the second largest l ...
) 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 Verbania (, , ) is the most populous ''comune'' (municipality) and the capital city of the province of Verbano-Cusio-Ossola in the Piedmont region of northwest Italy. It is situated on the shore of Lake Maggiore, about north-west of Milan and ab ...
(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 Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
,
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, and complex manifold theory. The theory of
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s was elaborated by
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and is still being applied in novel ways to
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
. In 1853,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
asked Riemann, his student, to prepare a ''
Habilitationsschrift Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including ...
'' on the foundations of geometry. Over many months, Riemann developed his theory of
higher dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
and delivered his lecture at Göttingen in 1854 entitled ''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''. It was not published until 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. The subject founded by this work is
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
. Riemann found the correct way to extend into ''n'' dimensions the
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
of surfaces, which Gauss himself proved in his ''
theorema egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
''. The fundamental objects are called the
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
and 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. ...
. For the surface (two-dimensional) case, the curvature at each point can be reduced to a number (scalar), with the surfaces of constant positive or negative curvature being models of the
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
. The Riemann metric is a collection of numbers at every point in space (i.e., a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, no matter how distorted it is.


Complex analysis

In his dissertation, he established a geometric foundation for
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
through
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, through which multi-valued functions like the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
(with infinitely many sheets) or the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
(with two sheets) could become
one-to-one function In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
s. Complex functions are
harmonic functions In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \fr ...
(that is, they satisfy Laplace's equation and thus the
Cauchy–Riemann equations 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 differenti ...
) on these surfaces and are described by the location of their singularities and the topology of the surfaces. The topological "genus" of the Riemann surfaces is given by g=w/2-n+1, where the surface has n leaves coming together at w branch points. For g>1 the Riemann surface has (3g-3) parameters (the " moduli"). His contributions to this area are numerous. The famous
Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...
says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either \mathbb or to the interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
, which was proved in the 19th century by Henri Poincaré and
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
. Here, too, rigorous proofs were first given after the development of richer mathematical tools (in this case, topology). For the proof of the existence of functions on Riemann surfaces he used a minimality condition, which he called the
Dirichlet principle In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. Formal statement Dirichlet's principle states that, if the functio ...
.
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
found a gap in the proof: Riemann had not noticed that his working assumption (that the minimum existed) might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed. Through the work of David Hilbert in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass was very impressed with Riemann, especially with his theory of
abelian function In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
s. When Riemann's work appeared, Weierstrass withdrew his paper from '' Crelle's Journal'' and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859. Weierstrass encouraged his student
Hermann Amandus Schwarz Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kumme ...
to find alternatives to the Dirichlet principle in complex analysis, in which he was successful. An anecdote from Arnold Sommerfeld shows the difficulties which contemporary mathematicians had with Riemann's new ideas. In 1870, Weierstrass had taken Riemann's dissertation with him on a holiday to Rigi and complained that it was hard to understand. The physicist
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...
assisted him in the work over night and returned with the comment that it was "natural" and "very understandable". Other highlights include his work on abelian functions and
theta functions In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum fiel ...
on Riemann surfaces. Riemann had been in a competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of
elliptic integrals In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
. Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated period matrices and characterized them through the "Riemannian period relations" (symmetric, real part negative). By Ferdinand Georg Frobenius and
Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
the validity of this relation is equivalent with the embedding of \mathbb^n/\Omega (where \Omega is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of n, this is the
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
of the Riemann surface, an example of an abelian manifold. Many mathematicians such as
Alfred Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
(Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on the zeros and poles) of a Riemann surface. According to Detlef Laugwitz, Detlef Laugwitz: ''Bernhard Riemann 1826–1866''. Birkhäuser, Basel 1996,
automorphic function In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. Factor ...
s 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 surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s.


Real analysis

In the field of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, he discovered the
Riemann integral 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. It was presented to the faculty at the University of G ...
in his habilitation. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the
Stieltjes integral Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at ...
goes back to the Göttinger mathematician, and so they are named together the
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
. 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 In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an ''L''1 function vanishes at infinity. It is of importance in harmonic analysis and asymptot ...
: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large ''n''. Riemann's essay was also the starting point for
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
's work with Fourier series, which was the impetus for
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
. 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 In mathematics, and particularly ordinary differential equations (ODEs), a monodromy matrix is the fundamental matrix (linear differential equation), fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. I ...
). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems.


Number theory

Riemann made some famous contributions to modern
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
. In a single short paper, the only one he published on the subject of number theory, he investigated the
zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * ...
that now bears his name, establishing its importance for understanding the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. The
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
was one of a series of conjectures he made about the function's properties. In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
), behind which a theta function lies. 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 \pi(x). Riemann knew of
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...
's work on the
Prime Number Theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
. He had visited Dirichlet in 1852.


Writings

Riemann's works include: * 1851 – '' Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse'', Inauguraldissertation, Göttingen, 1851. * 1857 – '' Theorie der Abelschen Functionen'', Journal fur die reine und angewandte Mathematik, Bd. 54. S. 101–155. * 1859 – ''Über die Anzahl der Primzahlen unter einer gegebenen Größe'', in: ''Monatsberichte der Preußischen Akademie der Wissenschaften.'' Berlin, November 1859, S. 671ff. With Riemann's conjecture. '' Über die Anzahl der Primzahlen unter einer gegebenen Grösse.'' (Wikisource)
Facsimile of the manuscript
with Clay Mathematics. * 1867 – '' Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe'', Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. * 1868
''Über die Hypothesen, welche der Geometrie zugrunde liegen''.
Abh. Kgl. Ges. Wiss., Göttingen 1868. Translatio
EMIS, pdf
'On the hypotheses which lie at the foundation of geometry'', translated by W.K.Clifford, Nature 8 1873 183 – reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea) http://www.emis.de/classics/Riemann/. Also in Ewald, William B., ed., 1996 “From Kant to Hilbert: A Source Book in the Foundations of Mathematics”, 2 vols. Oxford Uni. Press: 652–61. * 1876 – ''Berhard Riemann´s Gesammelte Mathematische Werke und wissenschaftlicher Nachlass. herausgegeben von Heinrich Weber unter Mitwirkung von Richard Dedekind'', Leipzig, B. G. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Max Noether and Wilhelm Wirtinger, Teubner 1902). Later editions ''The collected works of Bernhard Riemann: the complete German texts. Eds. Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., 1953, 1981, 2017 * 1876 – ''Schwere, Elektrizität und Magnetismus'', Hannover: Karl Hattendorff. * 1882 – ''Vorlesungen über Partielle Differentialgleichungen'' 3. Auflage. Braunschweig 1882. * 1901 – ''Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen''. PDF on Wikimedia Commons. On archive.org: * 2004 –


See also

* List of things named after Bernhard Riemann *
Non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
*
On the Number of Primes Less Than a Given Magnitude " die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is seminal9-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte der K ...
, Riemann's 1859 paper introducing the complex zeta function


References


Further reading

* . * . *


External links

*
The Mathematical Papers of Georg Friedrich Bernhard Riemann

Riemann's publications at emis.de
*

* ttps://web.archive.org/web/20160318034045/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ Bernhard Riemann's inaugural lecture*
Richard Dedekind (1892), Transcripted by D. R. Wilkins, Riemanns biography.
{{DEFAULTSORT:Riemann, Georg Friedrich Bernhard 1826 births 1866 deaths 19th-century deaths from tuberculosis 19th-century German mathematicians Differential geometers Foreign Members of the Royal Society German Lutherans Tuberculosis deaths in Italy People from the Kingdom of Hanover University of Göttingen alumni University of Göttingen faculty Infectious disease deaths in Piedmont