# Bernhard Riemann

Bernhard Riemann, engraving by August Weger (1863)

Georg Friedrich Bernhard Riemann (born September 17, 1826 in Breselenz near Dannenberg (Elbe) ; † July 20, 1866 in Selasca near Verbania on Lake Maggiore ) was a German mathematician who, despite his relatively short life, worked in many areas of analysis , differential geometry , mathematical physics and analytical number theory had a pioneering effect. He is considered one of the most important mathematicians.

## Life

### Origin and youth

Johanneum in Lüneburg 1829

Riemann grew up as one of five children in cramped conditions in a Lutheran rectory . His mother, the daughter of Hofrat Ebell in Hanover , died early (1846). His father, Friedrich Bernhard Riemann, who came from Boizenburg , had taken part in the wars of liberation (Army von Wallmoden ) and was last pastor in Quickborn . Riemann always kept close ties to his family.

From 1840 to 1842 he attended the grammar school in Hanover, then until 1846 the grammar school Johanneum in Lüneburg, from where he could watch the catastrophic fire of Hamburg in the distance. His math skills were noticed early on. A teacher, the rector Schmalfuss, lent him Legendre's number theory ( Théorie des Nombres ), a difficult work of 859 quarto- size pages, but got it back a week later and found Riemann to go far beyond the usual when he finished his Abitur checked that Riemann had made this book his own completely.

### Education

Bernhard Riemann as a student

Riemann was to become a theologian like his father and had already learned Hebrew in addition to Latin and Greek in Lüneburg; but then he switched to mathematics in Göttingen . From 1846 to 1847 he studied in Göttingen a. a. with Moritz Stern , Johann Benedict Listing - a pioneer of topology (he wrote a book about it in 1847) - and Carl Friedrich Gauß , who at that time read almost exclusively about astronomy and only rarely read about applied topics such as his least squares method .

1847–1849 Riemann heard lectures from Peter Gustav Dirichlet on partial differential equations in Berlin , from Jacobi and Gotthold Eisenstein - with whom he became closer - on elliptical functions , from Steiner geometry. After Richard Dedekind, he was also impressed by the events of the revolution of March 1848 - as part of the student corps, he kept watch in front of the royal palace for a day.

In 1849 he was back in Göttingen and began working on his dissertation with Gauss on function theory , which he completed in 1851. He then became a temporary assistant to the physicist Wilhelm Eduard Weber . In 1854 he completed his habilitation. The subject of his habilitation lecture on June 10, 1854 was About the hypotheses on which geometry is based . In 1855 his father died.

### Professor in Göttingen, Travel and Death

Elise Riemann b. cook

From 1857 onwards, Riemann held an extraordinary professorship in Göttingen. In the same year, his two remaining sisters moved in with him, for whom he had to look after his brother's death despite his low salary - at that time, a professor's salary consisted largely of student fees , and the more demanding the lecture, the fewer listeners usually appeared. Riemann suffered a breakdown from overwork and went to Dedekind to relax in Bad Harzburg . In 1858 the Italian mathematicians Francesco Brioschi , Enrico Betti and Felice Casorati visited him in Göttingen, with whom he befriended and whom he conveyed topological ideas. In the same year he visited Berlin again and met Ernst Eduard Kummer , Karl Weierstrass and Leopold Kronecker there . In 1859 he succeeded Dirichlet on the chair of Gauß in Göttingen. This marked a brief period of contentment in Riemann's life. His professorial salary lifted him from the poverty of his student years, and so he was eventually able to afford decent housing and even housekeeping. In 1860 he traveled to Paris and met Victor Puiseux , Joseph Bertrand , Charles Hermite , Charles Briot and Jean-Claude Bouquet .

In 1862 he married Elise Koch, a friend of his sisters, with whom he had a daughter, Ida, who was born in Pisa in 1863 . He then stayed longer in Italy and met his Italian mathematician friends again. On returning from a trip to Italy in 1862, his health deteriorated. Riemann suffered from tuberculosis . Even longer stays in the mild climate of Italy could not cure the disease. While searching for relaxation again on his third trip to Italy, he died at the age of 39 in Selasca on Lake Maggiore . He was buried in Biganzolo . The grave no longer exists, only the gravestone in the cemetery wall has been preserved.

His daughter Ida (1863–1929) was married to the mathematician and navigator Carl Schilling , and the widow Elise Riemann (1835–1904) and her sister Ida Koch (1825–1899) moved in 1890 to the Schillings in Bremen.

## plant

Despite his relatively short life, Riemann became one of the most outstanding mathematicians, whose work is of great importance for the natural sciences to this day. On the one hand, he was one of the founders of function theory , the theory of the functions of a complex variable. On the other hand, as the founder of Riemannian geometry, he is one of the pioneers of Einstein's general theory of relativity .

### geometry

He published his ideas on "Riemannian geometry", i. H. Differential geometry in any number of dimensions with locally defined metrics, only in his habilitation lecture in 1854, which he gave in the presence of the deeply impressed Carl Friedrich Gauß . He had proposed several topics and only listed the "Hypotheses Underlying Geometry" last. Gauss specifically chose this topic (which is actually unusual). In the lecture, Riemann was forced to express himself in a way that was understandable for a broader group, and therefore only a few formulas appear in it. In a Paris price publication (published in the Gesammelte Werken in 1876), Riemann indicated the more concrete implementation of his ideas (including Christoffel symbols , curvature tensor ).

### Function theory

His geometrical justification of the theory of functions with the introduction of Riemannian surfaces , on which ambiguous functions such as the logarithm (infinite number of leaves) or the root function (two leaves) become "unambiguous", happened in his dissertation, which, according to Dedekind, was completed in Berlin in autumn 1847 was (in discussions with Eisenstein he is said to have represented his differential equation approach to function theory compared to Eisenstein's more formal approach). Complex functions are " harmonic functions " (that is, they fulfill the Laplace equation or, equivalently, the Cauchy-Riemann differential equations ) on these surfaces and are described by the position of their singularities and the topology of these surfaces (number of cuts, etc.). The topological “gender” of the Riemann surfaces is given by, whereby leaves are attached to each other at the branch points of the surface . For the Riemann surface has parameters (the "modules"). ${\ displaystyle g = w / 2-n + 1}$${\ displaystyle w}$${\ displaystyle n}$${\ displaystyle g> 1}$${\ displaystyle (3g-3)}$

His contributions to this area are numerous. His famous Riemannian mapping theorem states that every simply connected area in the complex number plane C is equivalent to either the whole of C or the interior of the unit circle "biholomorphic" (that is, there is an analytic mapping, also in the opposite direction). The generalization of the theorem in relation to Riemann surfaces is the famous uniformization theorem , around which in the 19th century a. a. Henri Poincaré and Felix Klein tried hard. Here, too, strict proofs have only been given with the development of sufficient mathematical tools - in this case from the topology.

To prove the existence of functions on Riemann surfaces, he used a minimal condition that he called the Dirichlet principle . Karl Weierstrass immediately pointed out a loophole: With his “working hypothesis” (for him the existence of the minimum was clearly clear), Riemann failed to take into account that the underlying functional space need 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 put on theoretically secure ground around the turn of the century.

Weierstrass was also very impressed by Riemann, especially by his theory of Abelian functions. When this appeared, he withdrew his own manuscript, which was already with Crelle , and no longer published it. Both got along well when Riemann visited him in Berlin in 1859. Weierstrass encouraged his student Hermann Amandus Schwarz to look for alternatives to the Dirichlet principle in the foundation of the theory of functions, in which this was also successful. An anecdote passed down by Arnold Sommerfeld is indicative of the difficulties that contemporary mathematicians had with Riemann's new ideas : Weierstrass took Riemann's dissertation with him on vacation on the Rigi to study in the 1870s and complained that it was difficult to understand. The physicist Hermann von Helmholtz borrowed the work overnight and returned it with the comment that it was "natural" and "as a matter of course" for him.

Riemann

Further highlights are his work on Abelian functions and theta functions on Riemannian surfaces. Since 1857 Riemann was in a competition with Weierstrass to solve the Jacobian inversion problem of the Abelian integrals , a generalization of the elliptic integrals. Riemann used theta functions in several variables and reduced the problem to determining the zeros of these theta functions. Riemann also examined the period matrix (the G abelian integrals 1st genus on g paths, which result from the “canonical division” of the surface with 2g paths) and characterized it by the “Riemann period relations” (symmetrical, real part negative). According to Ferdinand Georg Frobenius and Solomon Lefschetz, the validity of these relations is equivalent to embedding , ( = grid from the period matrix) in a projective space by means of theta functions. For n = g this is the Jacobi variety of the Riemann surface also examined by Riemann, an example of an Abelian manifold (lattice). ${\ displaystyle C ^ {n} / \ Omega}$${\ displaystyle \ Omega}$

Numerous mathematicians such as B. Alfred Clebsch elaborated on Riemann's relationship to the theory of algebraic curves. This theory can be expressed by the properties of the functions that can be defined on a Riemann surface. For example, the Riemann-Roch theorem ( Roch was a Riemann student) makes statements about the number of linearly independent differentials (with certain specifications at their zero and pole positions) on a Riemann surface.

According to Laugwitz, automorphic functions appear for the first time in an essay on the Laplace equation on electrically conductive cylinders. However, Riemann also used such functions for conforming images, e.g. B. from circular arc triangles to the circle in his lectures on hypergeometric functions in 1859 (rediscovered by Schwarz) or in the treatise on minimal areas. Freudenthal sees it as Riemann's greatest mistake that he did not already allow Möbius transformations in his introduction of the Riemann surfaces to the sections and thus introduce automorphic functions (which he does at the singular points in the theory of the hypergeometric differential equation). Riemann knew the Gauss estate, in which the modular figure also appears.

### Number theory

His work on the number of prime numbers below a given size from 1859, his only work on number theory, is considered the founding text of analytic number theory, along with some works by Pafnuti Lwowitsch Tschebyschow and his teacher Dirichlet. It was about the attempt to prove and tighten the prime number theorem assumed by Gauss.

In this work he made very extensive statements about the distribution of prime numbers with the help of function theory . Above all, the Riemann hypothesis, named after him, about the zeros of the zeta function can be found here , but only mentioned in one sentence (he gave up the proof after a few fleeting attempts because it was not necessary for the immediate purpose of the treatise). It is of fundamental importance for number theory , but has not yet been proven. In 1932, when Siegel examined Riemann's estate in Göttingen , Siegel showed that behind this short essay there are also much more extensive calculations by Riemann.

There are many other interesting developments in Riemann's work. In this way he proved the functional equation of the zeta function (already known to Euler), behind which there is one of the theta function. It also gives a much better approximation for the prime number distribution than the Gaussian function Li ( x ). By summing this approximation function over the nontrivial zeros on the straight line with real part 1/2, he even gives an exact “explicit formula” for . ${\ displaystyle \ pi (x)}$${\ displaystyle \ pi (x)}$

Riemann was familiar with Chebyshev's work on the prime number theorem. He had visited Dirichlet in 1852. Riemann's methods are completely different.

### Real functions, Fourier series, Riemann integral, hypergeometric differential equation

In the field of real functions , he developed the Riemann integral, also named after him (in his habilitation). Among other things, he proved that every piecewise continuous function can be integrated. Likewise, the Stieltjes integral goes back to the Göttingen mathematician and is therefore sometimes also referred to as the Riemann- Stieltjes integral .

In his habilitation thesis on Fourier series , where he also followed in the footsteps of his teacher Dirichlet, he proved that Riemann-integrable functions can be "represented" by Fourier series. Dirichlet had proven this for continuous, piece-wise differentiable functions (i.e. with countably many jump points). As a case not covered by Dirichlet, Riemann gave the example of a continuous function, almost nowhere differentiable, in the form of a Fourier series. He also proved the Riemann-Lebesgue lemma : if a function can be represented by a Fourier series, the Fourier coefficients approach zero for large n .

Riemann's essay was also the starting point of Georg Cantor's study of Fourier series, from which set theory emerged.

He also treated the hypergeometric differential equation in 1857 with methods of function theory and characterized the solutions by the behavior described in the monodrome matrix on closed paths around the singularities. The proof of the existence of such a differential equation for a given monodrome matrix is ​​one of the Hilbert problems (Riemann-Hilbert problem).

Riemann in Florence, probably 1863

### Mathematical physics, natural philosophy

Riemann was also very interested in mathematical physics and natural philosophy under the influence of the philosopher Johann Friedrich Herbart . This represented a kind of "field theory" of mental phenomena similar to the electrodynamic in analogy to Gauss's theorem of potential theory. Herbart: "At every moment something permanent enters our soul, only to disappear again immediately." For Herbart, who sought a mathematical foundation for psychology by resorting to Hume , the subject was only the changeable product of ideas. As Riemann himself states, he was able to subscribe to some of Herbart's epistemological and psychological concepts, but not to his natural philosophy. His review of the early writings of Gustav Theodor Fechner shows that he shared Fechner's teaching influenced by Friedrich Wilhelm Joseph Schelling's natural philosophy, in particular the idea that there is an "inside of nature" that is animated by an "organizing principle and to" higher levels of development "leads. Riemann's ideas on natural philosophy from his estate are published in his collected works.

His “Contribution to Electrodynamics” of 1858, which he withdrew from the publication, was intended to standardize electrodynamics : Coulomb forces (gravity, electricity) from resistance to volume change, “electrodynamic” forces such as light, thermal radiation from resistance to change in length of a line element (he goes from Ampère's law of the interaction of two currents). Instead of Poisson's equation for the potential, he comes up with a wave equation with a constant speed of light. In developing his ideas he was influenced by Isaac Newton's 3rd Letter to Bentley (quoted in Brewster's "Life of Newton"). Rudolf Clausius found a serious mistake in the posthumously published work.

His use of the Dirichlet principle already indicates methods of variation, and Riemann has also written a work on minimal surfaces . After Laugwitz, Hattendorff, who published it posthumously, worked on it awkwardly and anticipated many of Hermann Amandus Schwarz's ideas .

In mathematical physics, for example, he worked on heat conduction problems, potential problems, hyperbolic differential equation (in 1860 he found a new method for solving differential equations describing shock waves) and figures of rotating liquids. The Riemann problem is named after him because of his investigations into hyperbolic equations . In the field of rotating fluids, he answered a question by Dirichlet and found new characters alongside those of Dedekind, Dirichlet and Colin MacLaurin, who were already familiar . He also looked at their stability ( anticipating Lyapunov ). Hattendorf published his lectures on partial differential equations in mathematical physics after his death. Later it became a well-known textbook in the editing of Heinrich Weber . Shortly before his death, he was working on a theory of the human ear.

### Effect and appreciation

Riemann's tombstone in Biganzolo, 2009

After his death in 1876, Riemann's friend Richard Dedekind , together with Heinrich Weber, published the first edition of his works (2nd edition 1892 by Heinrich Weber) (and provided a biography), including a lot of unpublished material (his housekeeper is expected to post further works shortly burned his death from ignorance). The popularization of his function theory, which at that time was in competition with the “power series” function theory à la Cauchy and Weierstrass, was mainly carried out by Felix Klein in his lectures in Leipzig and Göttingen, who did not shy away from emphasizing physical analogies. Even Carl Neumann contributed in various books to the spread of Riemann's ideas. This is why Riemann's function theory was successful with physicists like Hermann von Helmholtz from the start . Helmholtz applied it as early as 1868 in a work on the movement of liquids (conformal images) and in 1868, following on from Riemann, wrote a work on the later so-called "Riemann-Helmholtz spatial problem". For a long time, function theory remained suspect to mathematicians, not least thanks to Weierstrass' criticism of the Dirichlet principle.

In particular, Riemann's ideas fell on fertile ground in Italy, whose newly founded nation-state was very hungry for new ideas. Riemann, who enjoyed staying in Italy to restore his health, also had personal relationships with Italian mathematicians such as Enrico Betti and Eugenio Beltrami , and they even tried to pull him to Italy to a chair in Pisa. His illness and death prevented that.

His immediate German pupils included Friedrich Schottky , Gustav Roch (who died in the same year as Riemann and also of tuberculosis ), Friedrich Prym , who, like Roch, heard from Riemann in 1861 and immediately applied his methods to Kummer in his dissertation in 1862 .

Typical for Riemann was a conceptual thinking that connected many areas, but he was also very strong “technically”. Like his role model Dirichlet, however, he avoided invoices whenever possible. With him, topology began to play a central role in mathematics.

## various

The scientific legacy of Riemann is kept by the Central Archives of German Mathematicians' Legacies at the Lower Saxony State and University Library in Göttingen . It does not include any private letters or documents that remained in the family's hands. Some of the private letters from the possession of Erich Bessel-Hagen (who probably acquired them around the time of the Second World War) came to the Berlin State Library .

In his place of birth, Breselenz, the community of Jameln named a street after him, as did the cities of Berlin , Dannenberg (Elbe) , Göttingen , Jena , Leipzig and Lüneburg .

## Eponyms

The following mathematical structures are named after Riemann:

The following mathematical theorems are also named after Riemann:

• Riemann-Hurwitz formula , a relationship between branching order, number of leaves and gender in holomorphic images of compact Riemann surfaces
• Riemann's mapping theorem : every simply connected area can be mapped biholomorphically onto the open unit disk
• Riemann's theorem of liftability : a singularity of a holomorphic function can be corrected if and only if the function is restricted in a region around the singularity
• Riemann rearrangement theorem , a theorem about the rearrangement of conditionally convergent series
• Riemann-Roch theorem , a theorem about the number of independent meromorphic functions with given zeros and poles on a compact Riemann surface

The following are also named after Riemann:

## literature

• Eric Temple Bell : Men of mathematics . New York 1986 (first edition 1937). German under the title: The great mathematicians , Econ Verlag, 1967
• Umberto Bottazzini : Riemann's Influence on E. Betti and F. Casorati . In: Archive for History of Exact Sciences . Volume 18, No. 1, March 1977
• ders .: "Algebraic Truths" vs "Geometric Fantasies": Weierstrass' Response to Riemann . In: Proceedings of the International Congress of Mathematicians , Beijing, 20. – 28. August 2002
• Umberto Bottazzini and Rossana Tazzioli: “Natural philosophy and its role in Riemann's mathematics.” Revue d'Histoire des Mathématiques, Volume 1, 1995, pp. 3-38, numdam
• Umberto Bottazzini, Jeremy Gray : Hidden Harmony - Geometric Fantasies. The rise of complex function theory , Springer 2013
• Moritz CantorRiemann, Bernhard . In: Allgemeine Deutsche Biographie (ADB). Volume 28, Duncker & Humblot, Leipzig 1889, pp. 555-559.
• Richard Courant : Bernhard Riemann and the mathematics of the last 100 years , Natural Sciences, Volume 14, 1926, pp. 813–818, 1265–1277
• Olivier Darrigol : The mystery of Riemann's Curvature , Historia Mathematica, Volume 42, 2015, pp. 47-83
• Richard Dedekind : Bernhard Riemanns's curriculum vitae . In: Richard Dedekind, Heinrich Weber (ed.): Bernhard Riemann's collected mathematical works and scientific legacy. 2nd edition, Leipzig 1892, pp. 541–558, full text (PDF; 379 kB) at Heidelberg University
• John Derbyshire: Prime Obsession. Bernhard Riemann And The Greatest Unsolved Problem In Mathematics . Washington DC 2003, ISBN 0-309-08549-7
• Harold Edwards : Riemann's Zeta Function . Mineola, New York 2001 (Reprint), ISBN 0-486-41740-9
• Hans Freudenthal : Riemann . In: Dictionary of Scientific Biography . Vol. 11th Ed. Charles Coulston Gillipsie. New York: Scribner, 1975. 447-56.
• Lizhen Ji, Athanase Papadopoulos, Sumio Yamada (eds.): From Riemann to Differential Geometry and Relativity , Springer, 2017, XXXIV, ISBN 978-3-319-60039-0 (including introduction by Athanase Papadopoulos Looking backward: From Euler to Riemann )
• Felix Klein : Lectures on the development of mathematics in the 19th century . Springer-Verlag 1926, 1979.
• Detlef Laugwitz : Bernhard Riemann 1826-1866 . Birkhäuser, Basel 1996, ISBN 978-3-7643-5189-2
• Krzysztof Maurin: The Riemann legacy. Riemannian Ideas in Mathematics and Physics . Kluwer 1997
• Michael Monastyrsky: Riemann, Topology and Physics . 2nd Edition. Birkhäuser, 1999, ISBN 0-8176-3789-3
• Erwin Neuenschwander : Riemann and the “Weierstrasse” principle of analytical continuation through power series . Annual report of the German Mathematicians Association, Vol. 82, pp. 1–11 (1980)
• Neuenschwander: Lettres de Bernhard Riemann à sa famille . In: Cahiers du séminaire d'histoire des mathématiques , Vol. 2, 1981, pp. 85-131, numdam.org
• Olaf Neumann (Ed.): Bernhard Riemann (1826-1866). With B. Riemann, habilitation lecture, Göttingen 1854 (first published in Göttingen 1867 / BG Teubner 1876); R. Dedekind: Bernhard Riemann's curriculum vitae, BG Teubner 1876; O. Neumann: About Riemann's habilitation lecture, EAGLE 2017 , Leipzig, Edition am Gutenbergplatz Leipzig, 2017, ISBN 978-3-95922-097-2 [1]
• Olaf Neumann (ed.): Bernhard Riemann / Hermann Minkowski, Riemannsche spaces and Minkowski world. With B. Riemann's habilitation lecture, Göttingen 1854, and D. Hilbert's memorial address to H. Minkowski, Göttingen 1909. With original works by B. Riemann, H. Minkowski, R. Dedekind, D. Hilbert and the essay Riemann written by O. Neumann, Minkowski and the concept of space , Leipzig, Edition am Gutenbergplatz Leipzig, 2012, ISBN 978-3-937219-14-1 [2]
• Winfried Scharlau (ed.): Richard Dedekind: 1831–1981, a tribute to his 150th birthday , Braunschweig, Vieweg, 1981, ISBN 3-528-08498-7 (here also from Dedekind zu Riemann some of what he said in his biography concealed in the collected works out of consideration for the widow)
• Ernst Schering : Speech in memory of Riemann from December 1, 1899 , in: Riemann, Bernhard: Collected mathematical works and scientific legacy. Published with the assistance of Richard Dedekind and Heinrich Weber , Second Edition, Leipzig 1892, Vol. 2
• Erhard Scholz: Herbart's influence on Bernhard Riemann , Historia Mathematica, Volume 9, 1982, pp. 413-440
• Carl Ludwig Siegel : Lectures on selected chapters of the theory of functions , Göttingen, o.J./1995, Vol. 1,2 (explanation of Riemann's work), available here: uni-math.gwdg.de
• ders .: About Riemann's estate on analytical number theory , source studies on the history of mathematics, astronomy and physics, Dept. B: Studies 2, (1932), pp. 45–80. (Also in Gesammelte Abhandlungen , Vol. 1, Springer-Verlag, Berlin and New York 1979, ISBN 978-3-540-09374-9 ).
• Peter Ullrich:  Riemann, Georg Friedrich Bernhard. In: New German Biography (NDB). Volume 21, Duncker & Humblot, Berlin 2003, ISBN 3-428-11202-4 , p. 591 f. ( Digitized version ).
• Annette Vogt : The Development of Modern Function Theory in the Work of B. Riemann (1826 - 1866) and K. Weierstrass (1815 - 1897): A Comparison of Their Thinking Style , 1986 DNB 870532820 (Dissertation A Universität Leipzig 1986, 111 pages).
• André Weil : Riemann, Betti and the birth of topology , in: Archive for History of Exact Sciences , Vol. 20, 1979, p. 91 and Vol. 21, 1980, p. 387 (including a letter from Bettis, in which he made a statement Riemanns reports that he had the idea for his cuts from a conversation with Gauss)
• Hermann Weyl : Explanations in his edition of Riemann: Hypotheses which underlie the geometry. Springer, Berlin 1919
• Hermann Weyl : Riemann's geometric ideas, their effects and their connection with group theory . Springer, 1988

## Fiction

• Atle Næss : The Riemann Hypothesis. On the beauty of prime numbers and the mystery of love . Piper, Munich 2007, ISBN 978-3-492-05110-1 (Norwegian original title: 'Roten av minus en' ['Root of minus one'], translated by Günther Frauenlob). Pocket edition also from Piper, Munich 2009, ISBN 978-3-492-25366-6

Commons : Bernhard Riemann  - Album with pictures, videos and audio files
Wikisource: Bernhard Riemann  - Sources and full texts

## References and comments

1. The (very positive) assessment by Gauß and others is printed in Reinhold Remmert The Riemann file No. 135 of the Philosophical Faculty of Georgia Augusta at Göttingen , Mathematical Intelligencer, 1993, No. 3, p. 44
2. From June 28th he lived in the Villa Pisoni in Selasca
3. Riemann's grave in Biganzolo (accessed August 12, 2010)
4. Derbyshire Prime Obsession , Joseph Henry Press, p. 364. Gravestone of the widow and sister of Riemann, the daughter of Carl Schilling and her five children in Bremen-Riensberg
5. It is only further known through the publication in the news of the Göttinger Akad.Wiss. 1868 by Dedekind
6. Sommerfeld "Lectures on theoretical physics", Vol. 2 (Mechanics of deformable media), Harri Deutsch, p. 124. Sommerfeld had the story from the Aachen professor of experimental physics Adolf Wüllner.
7. Erhard Scholz : Herbert's Influence on Bernhard Riemann . In: Historia Mathematica , Vol. 9, 1982, pp. 413-440
8. Quoted from the Riemann biography of Laugwitz
9. Riemann, Werke, 1876, p. 476
10. see Marie-Luise Heuser : Schelling's Concept of Self-Organization, In: R. Friedrich / A. Wunderlin (ed.): Evolution of dynamical structures in complex systems. Springer Proceedings in Physics, Berlin / Heidelberg / New York (Springer) 1992, pp. 395-415 on the Riemannian reception of Schelling's natural philosophy via Fechner.
11. Marcus du Sautoy, The Music of Prime Numbers. On the trail of the greatest puzzle in mathematics , Munich 2003, ISBN 3-423-34299-4 , page 130.
12. Erwin Neuenschwander A brief report on a number of recently discovered sets of notes of Riemann's lectures and on the transmission of the Riemann Nachlass , Historia Mathematica, 15, 1988, 101–113
13. Riemann - A network monitoring system. Accessed on February 9, 2018 (English).