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, structure, space, models, and change.
History
On ...
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, "Ma ...
(including creating the field 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 Diri ...
), and to the theory of
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, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
; 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 peopl ...
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 E ...
, 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 e ...
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 ...
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
Liège ( , , ; wa, Lîdje ; nl, Luik ; german: Lüttich ) is a major city and municipality of Wallonia and the capital of the Belgian province of Liège.
The city is situated in the valley of the Meuse, in the east of Belgium, not far fro ...
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, 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 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 languages, 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 ...
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 ye ...
.
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 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 ...
who was nominally a professor of
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
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. Si ...
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 n ...
and at the
University of Paris
, image_name = Coat of arms of the University of Paris.svg
, image_size = 150px
, caption = Coat of Arms
, latin_name = Universitas magistrorum et scholarium Parisiensis
, motto = ''Hic et ubique terrarum'' (Latin)
, mottoeng = Here and a ...
, 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'', 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
**Ge ...
, 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 for the case , brought him immediate fame, being the first advance in the theorem since Fermat'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 ...
'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 name ...
, 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 ...
, 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 experim ...
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
Frederick William III (german: Friedrich Wilhelm III.; 3 August 1770 – 7 June 1840) was King of Prussia from 16 November 1797 until his death in 1840. He was concurrently Elector of Brandenburg in the Holy Roman Empire until 6 August 1806, w ...
. Humboldt, planning to make
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitu ...
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. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
. 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 hono ...
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, 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 '' ...
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.
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 ...
, both outstanding musicians, and the painter
Wilhelm Hensel
Wilhelm Hensel (6 July 1794 – 26 November 1861) was a German painter, brother of Luise Hensel, husband to Fanny Mendelssohn, and brother-in-law to Felix Mendelssohn.
Life and career
Wilhelm Hensel was born on 6 July 1794 in the German to ...
(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 ...
and the youngest sister of
Felix Mendelssohn
Jakob Ludwig Felix Mendelssohn Bartholdy (3 February 18094 November 1847), born and widely known as Felix Mendelssohn, was a German composer, pianist, organist and conductor of the early Romantic period. Mendelssohn's compositions include sym ...
and
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
A Singspiel (; plural: ; ) is a form of German-language music drama, now regarded as a genre of opera. It is characterized by spoken dialogue, which is alternated with ensembles, songs, ballads, and aria
In music, an aria ( Italian: ; plur ...
''
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 university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911.
General information
The ori ...
in 1858.
Berlin (1826–1855)
As soon as he came to Berlin, Dirichlet applied to lecture at the
University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative ...
, and the Education Minister approved the transfer and in 1831 assigned him to the faculty of
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
. 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 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 the developm ...
. He advised the doctoral theses of several important German mathematicians, as Gotthold Eisenstein,
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, ...
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 provi ...
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
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. 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, 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
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned ...
, 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, 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 Europe ...
; 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 ...
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 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
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
. 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 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, "Ma ...
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 approxima ...
, 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, "Ma ...
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 ...
, using
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
concepts to tackle an algebraic problem and thus creating the branch 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 Diri ...
. 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) \ch ...
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 r ...
. 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_\,\ ...