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, mathematical structure, structure, space, Mathematica ...
analysis Analysis (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
,
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 integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
, and
differential geometry Differential geometry is a Mathematics, 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 ...
. In the field of real analysis, 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 (mathematics), function on an Interval (mathematics), interval. It was pre ...
, and his work on
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
. 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 surfaces, 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 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 List ...
, 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 Dirich ...
. 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 differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
. 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, 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 of the United Kingdom, George III to his Hanoverian territories after the Napoleonic Wars, Napoleo ...
. 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 First French Empire, French Empire and its allies, led by Napoleon, Napoleon I, against a fluctuating array of Coalition forces of the Napoleonic Wars, Europe ...
. 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 States of Germany, state of Lower Saxony. Its 535,932 (2021) inhabitants make it the List of cities in Germany by population, 13th-largest city in Germa ...
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 th ...
(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 sacredness, sacred in Christianity, Judaism, Samaritanism, and many other religions. The Bible is an anthologya compilation of ...
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 Christian theology is the theology of Christianity, Christian belief and practice. Such study concentrates primarily upon the texts of the Old Testament and of the New Testament, as well as on Christian tradition. Christian theology, theologian ...
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 Theology is the systematic study of the nature of the Divinity, divine and, more broadly, of religious belief. It is taught as an Discipline (academia), academic discipline, typically in universities and seminaries. It occupies itself with 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 In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'out ...
). 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 Berlin ( , ) is the capital and largest city of Germany ...
in 1847. During his time of study, Carl Gustav Jacob Jacobi,
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, 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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
'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 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 States of Germany, state of Lower Saxony. Its 535,932 (2021) inhabitants make it the List of cities in Germany by population, 13th-largest city in Germa ...
and
Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a Germans, 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 ...
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 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 Algebraic geometry is a branch of 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. ...
, and
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas (topology), atlas of chart (topology), charts to the open unit disc in \mathbb^n, such that the transition maps are Holomorphic function, holomorphic. ...
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 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'' 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''. 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. Riemann found the correct way to extend into ''n'' dimensions the
differential geometry Differential geometry is a Mathematics, 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 ...
of surfaces, which Gauss himself proved in his '' theorema egregium''. 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 manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
and the Riemann curvature tensor. 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 geo ...
. The Riemann metric is a collection of numbers at every point in space (i.e., a
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
) 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 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 i ...
, 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 surfaces, 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 of ...
(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 (algebra), square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots o ...
(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 (mathematics), function that maps Distinct (mathematics), distinct elements of its domain to distinct elements; that is, implies . (Equivale ...
s. 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. 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 $\left(3g-3\right)$ parameters (the " moduli"). His contributions to this area are numerous. The famous Riemann mapping theorem 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, which was proved in the 19th century by Henri Poincaré and Felix Klein. 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.
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 mathematical analysis, analysis". Despite leaving university without a degree, ...
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 functions. 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 to find alternatives to the Dirichlet principle in complex analysis, in which he was successful. An anecdote from
Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a Germans, German theoretical physicist who pioneered developments in atomic physics, atomic and quantum physics, and also educated and mentored many students for th ...
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 Association ...
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 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. 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 Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a Germans, German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for th ...
and Solomon Lefschetz 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 of the Riemann surface, an example of an abelian manifold. Many mathematicians such as Alfred Clebsch 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 (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 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.

# Real analysis

In the field of real analysis, 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 (mathematics), function on an Interval (mathematics), interval. It was pre ...
in his habilitation. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, and so they are named together the Riemann–Stieltjes integral. In his habilitation work on
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
, 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''. 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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, ...
's work with Fourier series, which was the impetus for
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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, ...
. 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.

# 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 Dirich ...
. In a single short paper, the only one he published on the subject of number theory, he investigated the zeta 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 wa ...
s. The
Riemann hypothesis In mathematics, the Riemann hypothesis is the 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 List ...
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 $\pi\left(x\right)$. Riemann knew of
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician A mathematician is someone who uses an extensive knowledge of mathe ...
's work on the
Prime Number Theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, 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 pr ...
. 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 –

* 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 geo ...
* On the Number of Primes Less Than a Given Magnitude, Riemann's 1859 paper introducing the complex zeta function

* . * . *