Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) 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 ...
who made deep contributions to
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
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 '' ...
and other topics in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
; he is credited with being one of the first mathematicians to give the modern formal definition of a function.
Although his surname is Lejeune Dirichlet, he is commonly referred to by his
mononym
A mononym is a name composed of only one word. An individual who is known and addressed by a mononym is a mononymous person. In some cases, a mononym selected by an individual may have originally been from a polynym, a word which refers to one o ...
Dirichlet, in particular for results named after him.
Biography
Early life (1805–1822)
Gustav Lejeune Dirichlet was born on 13 February 1805 in
Düren
Düren (; ripuarian: Düre) is a town in North Rhine-Westphalia, Germany, between Aachen and Cologne on the river Rur.
History
Roman era
The area of Düren was part of Gallia Belgica, more specifically the territory of the Eburones, a pe ...
, a town on the left bank of the
Rhine
The Rhine ; french: Rhin ; nl, Rijn ; wa, Rén ; li, Rien; rm, label=Sursilvan, Rein, rm, label=Sutsilvan and Surmiran, Ragn, rm, label=Rumantsch Grischun, Vallader and Puter, Rain; it, Reno ; gsw, Rhi(n), including in Alsatian dialect, Al ...
which at the time was part of the
First French Empire
The First French Empire, officially the French Republic, then the French Empire (; Latin: ) after 1809, also known as Napoleonic France, was the empire ruled by Napoleon Bonaparte, who established French hegemony over much of continental ...
, reverting to
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 ...
after the
Congress of Vienna
The Congress of Vienna (, ) of 1814–1815 was a series of international diplomatic meetings to discuss and agree upon a possible new layout of the European political and constitutional order after the downfall of the French Emperor Napoleon B ...
in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelette (or more likely Richelle), a small community north east of Liège in
Belgium
Belgium, ; french: Belgique ; german: Belgien officially the Kingdom of Belgium, is a country in Northwestern Europe. The country is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to ...
, from which his surname "Lejeune Dirichlet" ("",
French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
for "the youth from Richelette") was derived.
Although his family was not wealthy and he was the youngest of seven children, his parents supported his education. They enrolled him in an elementary school and then private school in hope that he would later become a merchant. The young Dirichlet, who showed a strong interest in mathematics before age 12, persuaded his parents to allow him to continue his studies. In 1817 they sent him to the under the care of
Peter Joseph Elvenich Peter Joseph Elvenich (29 January 1796 – 16 June 1886) was a German Catholic theologian and philosopher born in Embken, a village that today is part of Nideggen, North Rhine-Westphalia. He was a principal supporter and defender of Hermesianism, a ...
, a student his family knew. In 1820, Dirichlet moved to the Jesuit Gymnasium in
Cologne
Cologne ( ; german: Köln ; ksh, Kölle ) is the largest city of the German western state of North Rhine-Westphalia (NRW) and the fourth-most populous city of Germany with 1.1 million inhabitants in the city proper and 3.6 millio ...
, where his lessons with
Georg Ohm
Georg Simon Ohm (, ; 16 March 1789 – 6 July 1854) was a German physicist and mathematician. As a school teacher, Ohm began his research with the new electrochemical cell, invented by Italian scientist Alessandro Volta. Using equipment of his ...
helped widen his knowledge in mathematics. He left the gymnasium a year later with only a certificate, as his inability to speak fluent
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
prevented him from earning the
Abitur
''Abitur'' (), often shortened colloquially to ''Abi'', is a qualification granted at the end of secondary education in Germany. It is conferred on students who pass their final exams at the end of ISCED 3, usually after twelve or thirteen yea ...
.
Studies in Paris (1822–1826)
Dirichlet again persuaded his parents to provide further financial support for his studies in mathematics, against their wish for a career in law. As Germany provided little opportunity to study higher mathematics at the time, with only
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 ...
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 i ...
who was nominally a professor of
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and anyway disliked teaching, Dirichlet decided to go to
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. ...
in May 1822. There he attended classes at the
Collège de France
The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris ...
and at the
University of Paris
The University of Paris (french: link=no, Université de Paris), Metonymy, metonymically known as the Sorbonne (), was the leading university in Paris, France, active from 1150 to 1970, with the exception between 1793 and 1806 under the French Revo ...
, learning mathematics from
Hachette Hachette may refer to:
* Hachette (surname)
* Hachette (publisher), a French publisher, the imprint of Lagardère Publishing
** Hachette Book Group, the American subsidiary
** Hachette Distribution Services, the distribution arm
See also
* Hachett ...
among others, while undertaking private study of Gauss's ''
Disquisitiones Arithmeticae
The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'', a book he kept close for his entire life. In 1823 he was recommended to General Maximilien Foy, who hired him as a private tutor to teach his children
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
, the wage finally allowing Dirichlet to become independent from his parents' financial support.
His first original research, comprising part of a proof of
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have bee ...
for the case , brought him immediate fame, being the first advance in the theorem since
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
's own proof of the case and
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 ...
's proof for .
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are nam ...
, one of the referees, soon completed the proof for this case; Dirichlet completed his own proof a short time after Legendre, and a few years later produced a full proof for the case . In June 1825 he was accepted to lecture on his partial proof for the case at the
French Academy of Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at th ...
, an exceptional feat for a 20-year-old student with no degree. His lecture at the Academy had also put Dirichlet in close contact with Fourier and Poisson, who raised his interest in
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...
As General Foy died in November 1825 and he could not find any paying position in France, Dirichlet had to return to Prussia. Fourier and Poisson introduced him to
Alexander von Humboldt
Friedrich Wilhelm Heinrich Alexander von Humboldt (14 September 17696 May 1859) was a German polymath, geographer, naturalist, explorer, and proponent of Romantic philosophy and science. He was the younger brother of the Prussian minister, ...
, who had been called to join the court of King Friedrich Wilhelm III. Humboldt, planning to make
Berlin
Berlin is Capital of Germany, the capital and largest city of Germany, both by area and List of cities in Germany by population, by population. Its more than 3.85 million inhabitants make it the European Union's List of cities in the European U ...
a center of science and research, immediately offered his help to Dirichlet, sending letters in his favour to the Prussian government and to the
Prussian Academy of Sciences
The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
. Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an unusual amount of praise that "Dirichlet showed excellent talent". With the support of Humboldt and Gauss, Dirichlet was offered a teaching position at the
University of Breslau
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. ''University'' is derived from the Latin phrase ''universitas magistrorum et scholarium'', which ...
. However, as he had not passed a doctoral dissertation, he submitted his memoir on the Fermat theorem as a thesis to the
University of Bonn
The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine ...
. Again his lack of fluency in Latin rendered him unable to hold the required public disputation of his thesis; after much discussion, the university decided to bypass the problem by awarding him an
honorary doctorate
An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or '' ad ho ...
in February 1827. Also, the Minister of Education granted him a dispensation for the Latin disputation required for the
Habilitation
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 ...
. Dirichlet earned the Habilitation and lectured in the 1827–28 year as a
Privatdozent
''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualific ...
at Breslau.
While in Breslau, Dirichlet continued his number theoretic research, publishing important contributions to the biquadratic reciprocity law which at the time was a focal point of Gauss's research. Alexander von Humboldt took advantage of these new results, which had also drawn enthusiastic praise from
Friedrich Bessel
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the met ...
, to arrange for him the desired transfer to Berlin. Given Dirichlet's young age (he was 23 years old at the time), Humboldt was able to get him only a trial position at the
Prussian Military Academy
The Prussian Staff College, also Prussian War College (german: Preußische Kriegsakademie) was the highest military facility of the Kingdom of Prussia to educate, train, and develop general staff officers.
Location
It originated with the ''A ...
in Berlin while remaining nominally employed by the University of Breslau. The probation was extended for three years until the position becoming definite in 1831.
Marriage to Rebecka Mendelssohn
230px, Dirichlet was married in 1832 to . They had two children, Walter (born 1833) and Flora (born 1845). Drawing by Rebecka Mendelssohn. They had two children, Walter (born 1833) and Flora (born 1845). Drawing by , 1823">Wilhelm Hensel">Rebecka Mendelssohn. They had two children, Walter (born 1833) and Flora (born 1845). Drawing by , 1823
After Dirichlet's move to Berlin, Humboldt introduced him to the great salons held by the banker Abraham Mendelssohn Bartholdy">Wilhelm Hensel, 1823
After Dirichlet's move to Berlin, Humboldt introduced him to the Salon (gathering)">great salons held by the banker Abraham Mendelssohn Bartholdy and his family. Their house was a weekly gathering point for Berlin artists and scientists, including Abraham's children Felix and Fanny Mendelssohn">Felix Mendelssohn">Felix and Fanny Mendelssohn, both outstanding musicians, and the painter Wilhelm Hensel (Fanny's husband). Dirichlet showed great interest in Abraham's daughter Rebecka, whom he married in 1832.
Rebecka Henriette Lejeune Dirichlet (née Rebecka Mendelssohn; 11 April 1811 – 1 December 1858) was a granddaughter of
Moses Mendelssohn
Moses Mendelssohn (6 September 1729 – 4 January 1786) was a German-Jewish philosopher and theologian. His writings and ideas on Jews and the Jewish religion and identity were a central element in the development of the ''Haskalah'', or ' ...
Fanny Mendelssohn
Fanny Mendelssohn (14 November 1805 – 14 May 1847) was a German composer and pianist of the early Romantic era who was also known as Fanny (Cäcilie) Mendelssohn Bartholdy and, after her marriage, Fanny Hensel (as well as Fanny Mendelssohn He ...
. Rebecka was born in
Hamburg
Hamburg (, ; nds, label=Hamburg German, Low Saxon, Hamborg ), officially the Free and Hanseatic City of Hamburg (german: Freie und Hansestadt Hamburg; nds, label=Low Saxon, Friee un Hansestadt Hamborg),. is the List of cities in Germany by popul ...
. In 1816 her parents arranged for her to be
baptised
Baptism (from grc-x-koine, βάπτισμα, váptisma) is a form of ritual purification—a characteristic of many religions throughout time and geography. In Christianity, it is a Christian sacrament of initiation and adoption, almost inv ...
at which point she took the names Rebecka Henriette Mendelssohn Bartholdy. She became a part of the notable ''
salon
Salon may refer to:
Common meanings
* Beauty salon, a venue for cosmetic treatments
* French term for a drawing room, an architectural space in a home
* Salon (gathering), a meeting for learning or enjoyment
Arts and entertainment
* Salon ...
'' of her parents,
Abraham Mendelssohn
Abraham Ernst Mendelssohn Bartholdy (born Abraham Mendelssohn; 10 December 1776 – 19 November 1835) was a German banker and philanthropist. He was the father of Fanny Mendelssohn, Felix Mendelssohn, Rebecka Mendelssohn, and Paul Mendelssohn.
...
and his wife Lea, having social contacts with the important musicians, artists and scientists in a highly creative period of German intellectual life. In 1829 she sang a small role in the premiere, given at the Mendelssohn house, of Felix's Singspiel ''
Die Heimkehr aus der Fremde
''Die Heimkehr aus der Fremde'' ( German, ''The Return Home from Abroad''), known in English as ''Son and Stranger'' or ''Return of the Roamer'',Mendelssohn family by
Alexander von Humboldt
Friedrich Wilhelm Heinrich Alexander von Humboldt (14 September 17696 May 1859) was a German polymath, geographer, naturalist, explorer, and proponent of Romantic philosophy and science. He was the younger brother of the Prussian minister, ...
. In 1833 their first son, Walter, was born. She died in
Göttingen
Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
in 1858.
Berlin (1826–1855)
As soon as he came to Berlin, Dirichlet applied to lecture at the
University of Berlin
The Humboldt University of Berlin (german: link=no, Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany.
The university was established by Frederick Will ...
, and the Education Minister approved the transfer and in 1831 assigned him to the faculty of philosophy. The faculty required him to undertake a renewed
habilitation
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 ...
qualification, and although Dirichlet wrote a ''Habilitationsschrift'' as needed, he postponed giving the mandatory lecture in Latin for another 20 years, until 1851. As he had not completed this formal requirement, he remained attached to the faculty with less than full rights, including restricted emoluments, forcing him to keep in parallel his teaching position at the Military School. In 1832 Dirichlet became a member of the
Prussian Academy of Sciences
The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
, the youngest member at only 27 years old.
Dirichlet had a good reputation with students for the clarity of his explanations and enjoyed teaching, especially as his University lectures tended to be on the more advanced topics in which he was doing research: number theory (he was the first German professor to give lectures on number theory), analysis and
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 ...
. He advised the doctoral theses of several important German mathematicians, as
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 ...
,
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
Carl Wilhelm Borchardt
Carl Wilhelm Borchardt (22 February 1817 – 27 June 1880) was a German mathematician.
Borchardt was born to a Jewish family in Berlin.
His father, Moritz, was a respected merchant, and his mother was Emma Heilborn. Borchardt studied und ...
, while being influential in the mathematical formation of many other scientists, including
Elwin Bruno Christoffel
Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later prov ...
Eduard Heine
Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician.
Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Le ...
Julius Weingarten
Julius Weingarten (2 March 1836 – 16 June 1910) was a German mathematician. He received his doctorate in 1864 from Martin-Luther-Universität Halle-Wittenberg.
He made some important contributions to the differential geometry of surfaces
...
. At the Military Academy, Dirichlet managed to introduce differential and
integral calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
in the curriculum, raising the level of scientific education there. However, he gradually started feeling that his double teaching load, at the Military academy and at the university, was limiting the time available for his research.
While in Berlin, Dirichlet kept in contact with other mathematicians. In 1829, during a trip, he met Carl Jacobi, at the time professor of mathematics at
Königsberg University
Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was ...
. Over the years they kept meeting and corresponding on research matters, in time becoming close friends. In 1839, during a visit to Paris, Dirichlet met
Joseph Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
, the two mathematicians becoming friends, keeping in contact and even visiting each other with the families a few years later. In 1839, Jacobi sent Dirichlet a paper by
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
, at the time a schoolteacher. Realizing Kummer's potential, they helped him get elected in the Berlin Academy and, in 1842, obtained for him a full professor position at the University of Breslau. In 1840 Kummer married Ottilie Mendelssohn, a cousin of Rebecka's.
In 1843, when Jacobi fell ill, Dirichlet traveled to Königsberg to help him, then obtained for him the assistance of King Friedrich Wilhelm IV's personal physician. When the physician recommended that Jacobi spend some time in Italy, Dirichlet joined him on the trip together with his family. They were accompanied to Italy by
Ludwig Schläfli
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
, who came as a translator; as he was strongly interested in mathematics, both Dirichlet and Jacobi lectured to him during the trip, and he later became an important mathematician himself. The Dirichlet family extended their stay in Italy to 1845, their daughter Flora being born there. In 1844, Jacobi moved to Berlin as a royal pensioner, their friendship becoming even closer. In 1846, when the
Heidelberg University
}
Heidelberg University, officially the Ruprecht Karl University of Heidelberg, (german: Ruprecht-Karls-Universität Heidelberg; la, Universitas Ruperto Carola Heidelbergensis) is a public research university in Heidelberg, Baden-Württemberg, ...
tried to recruit Dirichlet, Jacobi provided von Humboldt the needed support to obtain a doubling of Dirichlet's pay at the university in order to keep him in Berlin; however, even then he was not paid a full professor wage and could not leave the Military Academy.
Holding liberal views, Dirichlet and his family supported the
1848 revolution
The Revolutions of 1848, known in some countries as the Springtime of the Peoples or the Springtime of Nations, were a series of political upheavals throughout Europe starting in 1848. It remains the most widespread revolutionary wave in Europea ...
; he even guarded with a rifle the palace of the Prince of Prussia. After the revolution failed, the Military Academy closed temporarily, causing him a large loss of income. When it reopened, the environment became more hostile to him, as officers he was teaching were expected to be loyal to the constituted government. Some of the press who had not sided with the revolution pointed him out, as well as Jacobi and other liberal professors, as "the red contingent of the staff".
In 1849 Dirichlet participated, together with his friend Jacobi, in the jubilee of Gauss's doctorate.
Göttingen (1855–1859)
Despite Dirichlet's expertise and the honours he received, and even though, by 1851, he had finally completed all formal requirements for a full professor, the issue of raising his pay at the university still dragged on and he was still unable to leave the Military Academy. In 1855, upon Gauss's death, 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 i ...
decided to call Dirichlet as his successor. Given the difficulties faced in Berlin, he decided to accept the offer and immediately moved to Göttingen with his family.
Kummer Kummer is a German surname. Notable people with the surname include:
*Bernhard Kummer (1897–1962), German Germanist
*Clare Kummer (1873—1958), American composer, lyricist and playwright
*Clarence Kummer (1899–1930), American jockey
* Christo ...
was called to assume his position as a professor of mathematics in Berlin.
Dirichlet enjoyed his time in Göttingen, as the lighter teaching load allowed him more time for research and he came into close contact with the new generation of researchers, especially
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
and Bernhard Riemann. After moving to Göttingen he was able to obtain a small annual stipend for Riemann to retain him in the teaching staff there. Dedekind, Riemann,
Moritz Cantor
Moritz Benedikt Cantor (23 August 1829 – 10 April 1920) was a German historian of mathematics.
Biography
Cantor was born at Mannheim. He came from a Sephardi Jewish family that had emigrated to the Netherlands from Portugal, another branch of ...
and Alfred Enneper, although they had all already earned their PhDs, attended Dirichlet's classes to study with him. Dedekind, who felt that there were gaps in his mathematics education, considered that the occasion to study with Dirichlet made him "a new human being". He later edited and published Dirichlet's lectures and other results in
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
under the title (''Lectures on Number Theory'').
In the summer of 1858, during a trip to
Montreux
Montreux (, , ; frp, Montrolx) is a Swiss municipality and town on the shoreline of Lake Geneva at the foot of the Alps. It belongs to the district of Riviera-Pays-d'Enhaut in the canton of Vaud in Switzerland, and has a population of approxi ...
, Dirichlet suffered a heart attack. On 5 May 1859, he died in Göttingen, several months after the death of his wife Rebecka. Dirichlet's brain is preserved in the department of physiology at the University of Göttingen, along with the brain of Gauss. The Academy in Berlin honored him with a formal memorial speech presented by Kummer in 1860, and later ordered the publication of his collected works edited by Kronecker and Lazarus Fuchs.
Mathematics research
Number theory
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
was Dirichlet's main research interest, a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837,
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is al ...
, using
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \c ...
s and
L-functions
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ...
. Also, in the article he noted the difference between the
absolute Absolute may refer to:
Companies
* Absolute Entertainment, a video game publisher
* Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK
* Absolute Software Corporation, specializes in security and data risk manag ...
and
conditional convergence In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Definition
More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if
\lim_\,\ ...
of series and its impact in what was later called the Riemann series theorem. In 1841, he generalized his arithmetic progressions theorem from integers to the ring of
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...