**Lejeune Dirichlet** is best known for his proof that in any arithmetic progression with first term coprime to the difference there are infinitely many primes.

- By the age of 16 Dirichlet had completed his school qualifications and was ready to enter university.
- However, the standards in German universities were not high at this time so Dirichlet decided to study in Paris.
- It is interesting to note that some years later the standards in German universities would become the best in the world and Dirichlet himself would play a hand in the transformation.
- Dirichlet set off for France carrying with him Gauss's Disquisitiones arithmeticae Ⓣ(Investigations in arithmetic) a work he treasured and kept constantly with him as others might do with the Bible.
- In Paris by May 1822, Dirichlet soon contracted smallpox.
- From the summer of 1823 Dirichlet was employed by General Maximilien Sébastien Foy, living in his house in Paris.
- Dirichlet's first paper was to bring him instant fame since it concerned the famous Fermat's Last Theorem.
- Dirichlet proved case 1 and presented his paper to the Paris Academy in July 1825.
- On 28 November 1825 General Foy died and Dirichlet decided to return to Germany.
- There was a problem for Dirichlet since in order to teach in a German university he needed an habilitation.
- Although Dirichlet could easily submit an habilitation thesis, this was not allowed since he did not hold a doctorate, nor could he speak Latin, a requirement in the early nineteenth century.
- The problem was nicely solved by the University of Cologne giving Dirichlet an honorary doctorate, thus allowing him to submit his habilitation thesis on polynomials with a special class of prime divisors to the University of Breslau.
- From 1827 Dirichlet taught at Breslau but Dirichlet encountered the same problem which made him choose Paris for his own education, namely that the standards at the university were low.
- The Military College was not the attraction, of course, rather it was that Dirichlet had an agreement that he would be able to teach at the University of Berlin.
- Dirichlet had a lifelong friend in Jacobi, who taught at Königsberg, and the two exerted considerable influence on each other in their researches in number theory.
- However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV.
- Dirichlet then made a request for assistance from Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, which was successful.
- Dirichlet obtained leave of absence from Berlin for eighteen months and in the autumn of 1843 set off for Italy with Jacobi and Borchardt.
- Schläfli and Steiner were also with them, Schläfli's main task being to act as their interpreter but he studied mathematics with Dirichlet as his tutor.
- Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845.
- Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties.
- Dirichlet did not accept the offer from Göttingen immediately but used it to try to obtain better conditions in Berlin.
- The quieter life in Göttingen seemed to suit Dirichlet.
- We should now look at Dirichlet's remarkable contributions to mathematics.
- Shortly after publishing this paper Dirichlet published two further papers on analytic number theory, one in 1838 with the next in the following year.
- These papers introduce Dirichlet series and determine, among other things, the formula for the class number for quadratic forms.
- This work led him to the Dirichlet problem concerning harmonic functions with given boundary conditions.
- Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions.
- Dirichlet's work is published in Crelle's Journal in 1828.
- Because of this work Dirichlet is considered the founder of the theory of Fourier series.
- important parts of mathematics were influenced by Dirichlet.
- With Dirichlet began the golden age of mathematics in Berlin.

Born 13 February 1805, Düren, French Empire (now Germany). Died 5 May 1859, Göttingen, Hanover (now Germany).

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Algebra, Analysis, Origin Germany, Number Theory, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive