Peter Gustav Lejeune Dirichlet was a German mathematician. He made valuable contributions to number theory, analysis, and mechanics, and is credited with being one of the first mathematicians to give the modern formal definition of a function.

Background

Peter Gustav Lejeune Dirichlet was born on February 13, 1805, in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815. He was the youngest of seven children. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelle, a small community 5 km northeast of Liège in Belgium, from which his surname "Lejeune Dirichlet" was derived.

Education

Dirichlet first attended public school, then a private school that emphasized Latin. He was precociously interested in mathematics; it is said that before the age of twelve he used his pocket money to buy mathematical books. In 1817 he was sent to the Gymnasium in Bonn under the care of Peter Joseph Elvenich, a student his family knew. He is reported to have been an unusually attentive and well-behaved pupil who was particularly interested in modern history as well as in mathematics.

After two years in Bonn, Dirichlet was sent to a Jesuit college in Cologne that his parents preferred. Among his teachers was the physicist Georg Simon Ohm, who gave him a thorough grounding in theoretical physics. Dirichlet completed his Abitur examination at the very early age of sixteen. His parents wanted him to study law, but mathematics was already his chosen field.

Dirichlet arrived in Paris in May 1822. Shortly afterward he suffered an attack of smallpox, but it was not serious enough to interrupt for long his attendance at lectures at the Collège de France and the Faculté des Sciences.

In the summer of 1823, Dirichlet was fortunate in being appointed to a well-paid and pleasant position as tutor to the children of General Maximilien Fay, a national hero of the Napoleonic wars and then the liberal leader of the opposition in the Chamber of Deputies. He was treated as a member of the family and met many of the most prominent figures in French intellectual life. Among the mathematicians, he was particularly attracted to Fourier, whose ideas had a strong influence upon his later works on trigonometric series and mathematical physics.

Dirichlet’s first interest in mathematics was number theory. This interest had been awakened through an early study of Gauss’ famous Disquisitiones arithmeticae (1801), until then not completely understood by mathematicians. In June 1825 he presented to the French Academy of Sciences his first mathematical paper, “Mémoire sur l’impossibilité de quelques équations indéterminées du cinquième degré.”

General Fay died in November 1825, and the next year Dirichlet decided to return to Germany, a plan strongly supported by Alexander von Humboldt, who worked for the strengthening of the natural sciences in Germany. He was permitted to qualify for habilitation as Privatdozent at the University of Breslau; since he did not have the required doctorate, this was awarded honoris causa by the University of Cologne. His habilitation thesis dealt with polynomials whose prime divisors belong to special arithmetic series. A second paper from this period was inspired by Gauss’ announcements on the biquadratic law of reciprocity.

Dirichlet was appointed extraordinary professor in Breslau, but the conditions for scientific work were not inspiring. In 1828 he moved to Berlin, again with the assistance of Humboldt, to become a teacher of mathematics at the military academy. Shortly afterward, at the age of twenty-three, he was appointed extraordinary (later ordinary) professor at the University of Berlin.

Dirichlet’s first paper dealing with Fermat’s equation was inspired by Legendre; he returned only once to this problem, showing the impossibility of the case "n = 14." The subsequent number theory papers dating from the early years in Berlin were evidently influenced by Gauss and the Disquisitiones. Some of them were improvements on Gauss’ proofs and presentation, but gradually Dirichlet cut much deeper into the theory. There are papers on quadratic forms, the quadratic and biquadratic laws of reciprocity, and the number theory of fields of quadratic irrationalities, with the extensive discussion of the Gaussian integers "a + ib," where a and b are integers.

At a meeting of the Academy of Sciences on July 27, 1837, Dirichlet presented his first paper on analytic number theory. In this memoir he gives a proof of the fundamental theorem that bears his name: any arithmetical series of integers where a and b are relatively prime, must include an infinite number of primes. This result had long been conjectured and Legendre had expended considerable effort upon finding a proof, but it had been established only for a few special cases.

The paper on the primes in arithmetic progressions was followed in 1838 and 1839 by a two-part paper on analytic number theory, “Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres.” Dirichlet begins with a few general observations on the convergence of the series now called Dirichlet series. The main number theory achievement is the determination of the formula for the class number for quadratic forms with various applications. Also from this period are his studies on Gaussian sums.

These studies on quadratic forms with rational coefficients were continued in 1842 in an analogous paper on forms with coefficients that have Gaussian coefficients. It contains an attempt at a systematic theory of algebraic numbers when the prime factorization is unique, although it is restricted to Gaussian integers. It is of interest to note that here one finds the first application of Dirichlet’s Schubfachprinzip (“box principle”). This deceptively simple argument, which plays an important role in many arguments in modern number theory, may be stated as follows: if one distributes more than n objects in n boxes, then at least one box must contain more than one object.

It is evident from Dirichlet’s papers that he searched very intently for a general algebraic number theory valid for fields of arbitrary degree. He was aware of the fact that in such fields there may not be a unique prime factorization, but he did not succeed in creating a substitute for it: the ideal theory later created by Ernst Kummer and Richard Dedekind or the form theory of Leopold Kronecker.

In 1863, Dirichlet’s Vorlesungen über Zahlentheorie was published by his pupil and friend Richard Dedekind. To the later editions of this work Dedekind most appropriately added several supplements containing his own investigations on algebraic number theory. These addenda are considered one of the most important sources for the creation of the theory of ideals, which has now become the core of algebraic number theory.

Parallel with Dirichlet’s investigations on number theory was a series of studies on analysis and applied mathematics. His first papers on these topics appeared during his first years in Berlin and were inspired by the works of the French mathematicians whom he had met during his early years in Paris. His first paper on analysis is rather formal, generalizing certain definite integrals introduced by Laplace and Poisson. This paper was followed in the same year (1829) by a celebrated one published in Crelle’s Journal, as were most of his mathematical papers: “Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre deux limites données.” The paper was written under the influence of Fourier’s theory of heat conduction as presented in his Théorie analytique de la chaleur.

Dirichlet’s contributions to general mechanics began with three papers published in 1839. All three have nearly the some content; the most elaborate has the title “Über eine neue Methode zur Bestimmung vielfacher Integrale.” All deal with methods based upon a so-called discontinuity factor for evaluating multiple integrals, and they are applied particularly to the problem of determining the attraction of an ellipsoid upon an arbitrary mass point outside or inside the ellipsoid.

In the brief article “Über die Stabilität des Gleichgewichts” (1846), Dirichlet considers a general problem inspired by Laplace’s analysis of the stability of the solar system. He takes the general point of view that the particles attract or repel each other by forces depending only on the distance and acting along their central line; in addition, the relations connecting the coordinates shall not depend on time. Stability is defined as the property that the deviations of the coordinates and velocities from their initial values remain within fixed, small bounds. Dirichlet criticizes as unsatisfactory the previous analyses of the problem, particularly those by Lagrange and Poisson that depended upon infinite series expansions in which terms above the second order were disregarded without sufficient justification. Dirichlet avoids this pitfall by reasoning directly on the properties of the expression for the energy of the system.

Among the later papers on theoretical mechanics, one must mention “Über die Bewegung eines festen Körpers in einem incompressibeln flüssigen Medium” (1852), which deals with the motion of a sphere in an incompressible fluid; it is noteworthy for containing the first exact integration for the hydrodynamic equations. This subject occupied Dirichlet during his last years; in his final paper, “Untersuchungen über ein Problem der Hydrodynamik” (1857), he examines a related topic, but this includes only a minor part of his hydrodynamic theories. After his death, his notes on these subjects were edited and published by Dedekind in an extensive memoir.

Dirichlet was elected a member of the Prussian Academy of Sciences (1832), corresponding member of the Saint Petersburg Academy of Sciences (1833), member of the Göttingen Academy of Sciences (1846), foreign member of the French Academy of Sciences (1854), member of the Royal Swedish Academy of Sciences (1854), member of the Royal Belgian Academy of Sciences (1855), and foreign member of the Royal Society (1855).

Prussian Academy of Sciences
,
Germany

1832

Saint Petersburg Academy of Sciences
,
Russian Federation

1833

Göttingen Academy of Sciences
,
Germany

1846

French Academy of Sciences
,
France

1854

Royal Swedish Academy of Sciences
,
Sweden

1854

Royal Belgian Academy of Sciences
,
Belgium

1855

Royal Society
,
United Kingdom

1855

Personality

Dirichlet was an excellent teacher, always expressing himself with great clarity. His manner was modest; in his later years, he was shy and at times reserved. He seldom spoke at meetings and was reluctant to make public appearances. In many ways, he was a direct contrast to his lifelong friend, the mathematician Carl Gustav Jacobi.

Connections

Dirichlet married Rebecka Mendelssohn in 1832. They had four children, but only two of them, Walter and Florentina, survived.

In 1855 Dirichlet was awarded the civil class mIn 1855 Dirichlet was awarded the civil class medal of the Pour le Mérite order at von Humboldt's recommendation.