Peter Gustav Lejeune Dirichlet

How to use this course

Peter Gustav Lejeune Dirichlet published relatively little, and almost every paper changed its subject. He gave the first rigorous convergence theorem for Fourier series, proved that every suitable arithmetic progression contains infinitely many primes, computed class numbers of quadratic forms, found the structure of the unit group of a number field, and set the boundary problem for harmonic functions that bears his name. He was also the teacher who made the work of Carl Friedrich Gauss readable and who formed Leopold Kronecker, Gotthold Eisenstein, Bernhard Riemann and Richard Dedekind.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow his work roughly in order: analysis (1829), analytic number theory (1837–1840), approximation and units (1842–1846), potential theory (1850s) and average orders (1849).

Timeline

Düren, Bonn and Cologne, 1805–1822

Johann Peter Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town between Aachen and Cologne. The Rhineland was then part of Napoleon’s empire; after 1815 it became Prussian. His father, Johann Arnold Lejeune Dirichlet, was the town’s postmaster, a merchant and a city councillor. The unusual surname came from the family’s origins near Liège: “le jeune de Richelette”, the young man from Richelette, a village whose identification is uncertain.

The boy spent his pocket money on mathematics books. In 1817 he entered the Gymnasium in Bonn, and about three years later he moved to the Jesuit Gymnasium in Cologne. There Georg Simon Ohm, who later found the law of electrical resistance, taught him mathematics and widened his reading. Dirichlet left school at sixteen with a leaving certificate but without the Abitur. His spoken Latin was weak, a fact that would cause him trouble in German universities for decades.

German mathematics outside Göttingen was then thin, and Gauss in Göttingen hardly taught. Paris, with Laplace, Legendre, Fourier, Poisson and Cauchy, was the centre of the mathematical world. His parents agreed to let him go.

Paris, 1822–1826

Dirichlet arrived in Paris in May 1822 and attended lectures at the Collège de France and the Faculté des Sciences. Soon afterwards he fell ill with smallpox but recovered. In November 1822 his mother sent him a copy of Gauss’s Disquisitiones Arithmeticae. He studied it closely for the rest of his life, and it is said that he always kept it on his desk.

In the summer of 1823 he became tutor to the children of General Maximilien Sébastien Foy, a hero of the Napoleonic wars and a leader of the liberal opposition. The position gave him a home, a salary and an entry into Parisian society. In Foy’s house he met Joseph Fourier, whose theory of heat and trigonometric series shaped his analysis, and Siméon Denis Poisson.

Fermat’s equation for \(n=5\)

On 11 June 1825, at twenty, Dirichlet presented to the Academy of Sciences a memoir on the equation \(x^5+y^5=z^5\). Fermat had settled the exponent 4 and Euler the exponent 3. Dirichlet treated part of the case in which one of the unknowns is divisible by 5. Adrien-Marie Legendre, one of the referees, completed the argument soon afterwards, and the exponent 5 was settled by the two of them. The memoir made Dirichlet known. In 1832 he proved the case \(n=14\).

General Foy died in November 1825. Through Fourier, Dirichlet had met Alexander von Humboldt, who wanted to strengthen science in Prussia. Humboldt recommended him to the Prussian minister of education, and in 1826 Dirichlet returned home.

Breslau and Berlin, 1827–1843

To teach at a university Dirichlet needed a doctorate, and he could not hold the required Latin disputation. On 24 February 1827 the University of Bonn granted him an honorary doctorate. On his way east he visited Gauss in Göttingen on 18 March 1827, the start of a long friendship. He spent a year as a lecturer at Breslau, where his habilitation thesis dealt with polynomials whose prime divisors lie in arithmetic progressions.

In 1828 he moved to Berlin, again with Humboldt’s help, to teach at the Military School. He continued to teach there for 27 years. From 1831 he was also extraordinary professor at the university, and from 1839 ordinary professor. Because he had not given a Latin lecture, he did not obtain full rights in the faculty until 1851.

The Mendelssohn family

At the Berlin salons of the Mendelssohn family Dirichlet met Rebecka Mendelssohn, the sister of the composers Felix Mendelssohn Bartholdy and Fanny Hensel and a granddaughter of the philosopher Moses Mendelssohn. They married on 22 May 1832. Their son Walter was born in 1833; a son Ernst and a daughter Flora followed. The same year, aged 27, Dirichlet became the youngest member of the Prussian Academy of Sciences.

The 1830s were his most productive decade. His closest friend in Berlin and Königsberg was Carl Gustav Jacobi. The two were opposites: Jacobi quick and combative, Dirichlet slow, careful and reserved. Both worked at the meeting point of analysis and number theory, and Kummer, Kronecker and Eisenstein grew up in their circle.

Convergence of Fourier series

For the general reader. Fourier claimed that any function could be written as a sum of sines and cosines, but he never proved that the infinite sum actually adds up to the function. In 1829 Dirichlet gave the first proof, for a wide and precisely described class of functions. To do so he had to say exactly what a function is.

For a \(2\pi\)-periodic integrable function \(f\), the Fourier coefficients are \(\hat f(k)=\frac1{2\pi}\int_{-\pi}^{\pi}f(t)e^{-ikt}\,dt\) and the partial sums are \(S_Nf(x)=\sum_{|k|\le N}\hat f(k)e^{ikx}\). Substituting the coefficients gives

Proof sketch

Since \(\frac1{2\pi}\int_0^\pi D_N=\frac12\), it suffices to show \(\frac1\pi\int_0^\pi f(x-t)D_N(t)\,dt\to f(x^-)\) and its mirror image. On each piece \(f\) is monotone, and Dirichlet used a mean-value argument (Bonnet’s second mean-value theorem in later language) to control \(\int g(t)\frac{\sin(\lambda t)}{t}\,dt\) for monotone \(g\). The key limit is

Away from \(t=0\), the factor \(1/\sin(t/2)\) is bounded and the oscillation of \(\sin((N+\frac12)t)\) kills the integral (the Riemann–Lebesgue lemma).

What is a function?

In 1837 Dirichlet described a function of \(x\) on an interval as any rule assigning a single value \(y\) to each \(x\), with no need for a formula. In the 1829 paper he already gave the function equal to one constant on the rationals and another on the irrationals, as a function to which his integrals do not apply. Whether he was the first to state the general definition is debated by historians, but his usage set the modern standard.

Primes in arithmetic progressions

For the general reader. Are there infinitely many primes ending in 7? Euclid’s argument shows there are infinitely many primes, but says nothing about which kinds. In 1837 Dirichlet proved that any progression \(a,\ a+m,\ a+2m,\dots\) with no common factor between \(a\) and \(m\) contains infinitely many primes. His method, using functions of a continuous variable to count primes, created analytic number theory.

Characters and L-functions

A Dirichlet character mod \(m\) is a homomorphism \(\chi:(\mathbb Z/m)^\times\to\mathbb C^\times\), extended to \(\mathbb Z\) by \(\chi(n)=0\) when \(\gcd(n,m)>1\). There are \(\varphi(m)\) of them, and they satisfy the orthogonality relation

The L-function is \(L(s,\chi)=\sum_{n\ge1}\chi(n)n^{-s}\), absolutely convergent for \(s>1\). Since \(\chi\) is completely multiplicative, unique factorization gives the Euler product \(L(s,\chi)=\prod_p(1-\chi(p)p^{-s})^{-1}\), and taking logarithms,

The argument

Multiply by \(\overline{\chi(a)}\), sum over \(\chi\) and divide by \(\varphi(m)\):

For the principal character \(\chi_0\), \(L(s,\chi_0)\) differs from \(\zeta(s)\) by finitely many Euler factors, so \(\log L(s,\chi_0)\sim\log\frac1{s-1}\). For \(\chi\ne\chi_0\), the partial sums of \(\chi(n)\) are bounded, so \(L(s,\chi)\) extends continuously to \(s=1\). The theorem follows if \(L(1,\chi)\ne0\) for every \(\chi\ne\chi_0\).

For complex \(\chi\) the non-vanishing follows from \(\prod_\chi L(s,\chi)\ge1\) for \(s>1\): a zero at \(s=1\) of \(L(s,\chi)\) would be shared by \(L(s,\bar\chi)\), and two zeros would overwhelm the single pole of \(L(s,\chi_0)\). The hard case is a real character. Dirichlet handled it by recognizing \(L(1,\chi)\) as a class number of quadratic forms times a positive factor (Lesson 06), and a class number is at least 1.

The class number formula

For the general reader. Gauss had sorted quadratic forms \(ax^2+bxy+cy^2\) into finitely many classes but had no formula for how many there are. Dirichlet found one, expressing the count as the value of an L-function. It joined two worlds: the algebra of forms and the analysis of infinite series.

Let \(D\) be a fundamental discriminant, \(K=\mathbb Q(\sqrt D)\), and \(\chi_D(n)=\left(\frac Dn\right)\) the Kronecker symbol, a real primitive character mod \(|D|\). Let \(h(D)\) be the class number (of forms, or of ideals of the ring of integers of \(K\)).

Idea of proof

Count the pairs \((x,y)\) with \(f(x,y)\le N\), summed over a set of representatives \(f\) of the classes, in two ways. Geometrically, each form contributes the area of an ellipse, \(2\pi N/\sqrt{|D|}\), up to an error of lower order and the factor \(w\) for automorphisms. Arithmetically, the number of representations of \(n\) by all classes together is \(w\sum_{d\mid n}\chi_D(d)\). Summing over \(n\le N\) gives about \(wN\,L(1,\chi_D)\). Comparing the two counts gives the formula. For \(D>0\) the ellipses become hyperbolic regions, and one must cut out a fundamental domain for the action of the units, which produces \(\log\varepsilon\).

Because \(h(D)\ge1\) and the other factors are positive, \(L(1,\chi_D)>0\), the step missing in Lesson 05. Every real non-principal character is induced from some \(\chi_D\), so this covers all cases.

Pigeonholes and approximation

For the general reader. If you put more letters than pigeonholes, some hole gets two letters. Dirichlet turned this obvious remark, which he called the Schubfachprinzip (drawer principle), into a tool for proving that numbers exist without finding them. Its most famous use shows that every irrational number has unusually good fractional approximations.

Proof. Consider the \(N+1\) numbers \(\{k\alpha\}\), \(k=0,1,\dots,N\), in \([0,1)\), and the \(N\) boxes \([\frac jN,\frac{j+1}N)\). Two of them, \(\{k\alpha\}\) and \(\{l\alpha\}\) with \(k<l\), share a box. Then \(q=l-k\) and \(p=\lfloor l\alpha\rfloor-\lfloor k\alpha\rfloor\) work. If \(\alpha\) is irrational, \(|q\alpha-p|>0\), and choosing larger \(N\) produces new fractions without end. The same argument in \(n\) dimensions gives \(q\le N^n\) with \(\|q\alpha_i-p_i\|<1/N\) for all \(i\).

Pell’s equation

Sketch. For each of the infinitely many \(p/q\) with \(|p-q\sqrt d|<1/q\), we have \(|p^2-dq^2|=|p-q\sqrt d|\,|p+q\sqrt d|<2\sqrt d+1\). So some value \(k\) occurs infinitely often, and by pigeonholes again two solutions \((p_1,q_1),(p_2,q_2)\) of \(p^2-dq^2=k\) are congruent mod \(|k|\). Then \(\frac{p_1-q_1\sqrt d}{p_2-q_2\sqrt d}=x+y\sqrt d\) has integer \(x,y\), norm 1, and is not \(\pm1\).

The unit theorem

For the general reader. In ordinary integers the only numbers with an inverse are \(1\) and \(-1\). In richer number systems, such as numbers \(a+b\sqrt2\), there can be infinitely many: \(1+\sqrt2\) has inverse \(\sqrt2-1\). In 1846 Dirichlet determined the structure of these units in every number field. The answer depends only on how the field sits inside the real and complex numbers.

Let \(K\) be a number field of degree \(n=r_1+2r_2\), with \(r_1\) real embeddings and \(r_2\) pairs of complex embeddings, and let \(\mathcal O_K\) be its ring of integers.

Dirichlet proved it for rings \(\mathbb Z[\alpha]\) generated by an algebraic integer; the general form uses Dedekind’s ring \(\mathcal O_K\).

Proof sketch

Order the embeddings \(\sigma_1,\dots,\sigma_{r_1+r_2}\), one from each complex pair, and define \(\ell(u)=(\log|\sigma_i(u)|^{e_i})_i\in\mathbb R^{r_1+r_2}\), with \(e_i=1\) for real and \(2\) for complex embeddings. Then:

1. \(\ell\) is a homomorphism, and its kernel is finite (bounded algebraic integers of bounded degree are finite in number), hence equals \(\mu(K)\).
2. For a unit, \(|N(u)|=1\), so \(\ell(u)\) lies in the hyperplane \(\sum x_i=0\) of dimension \(r_1+r_2-1\).
3. \(\ell(\mathcal O_K^\times)\) is discrete, since only finitely many units have all \(|\sigma_i(u)|\) bounded.
4. The hard step: the image spans the hyperplane. This is where Dirichlet used the pigeonhole principle, constructing many algebraic integers of bounded norm with prescribed sizes of conjugates.

The Dirichlet problem and Dirichlet’s principle

For the general reader. If you fix the temperature along the edge of a metal plate and wait, the temperature inside settles into a steady state. Finding it is the Dirichlet problem. Dirichlet taught that the steady state is the one with the least “energy”, a principle Riemann used to build his theory of complex functions. Its justification became one of the dramas of nineteenth-century analysis.

By the maximum principle there is at most one solution. On the unit disc the solution is given by the Poisson integral

Gauss, William Thomson and Dirichlet used this reasoning in potential theory, and Dirichlet taught it in his Berlin lectures. Riemann named it Dirichlet’s principle in his 1857 memoir on Abelian functions and used it to prove the existence of functions on Riemann surfaces. The argument assumed that a minimizer exists because \(E\ge0\). Karl Weierstrass pointed out in 1870 that a positive functional need not attain its infimum. The principle was rescued by David Hilbert in 1900 and became, through Sobolev spaces and the direct method of the calculus of variations, a foundation of modern PDE theory.

Average orders and the divisor problem

For the general reader. The number of divisors of \(n\) jumps around wildly: 2 for a prime, many for a number like 720. Dirichlet showed in 1849 that on average it behaves very regularly, like \(\log n\), and found the next term in its growth with a simple geometric trick.

Let \(d(n)\) be the number of positive divisors of \(n\). Then \(\sum_{n\le x}d(n)\) counts lattice points \((a,b)\) with \(a,b\ge1\) and \(ab\le x\), the points under a hyperbola.

The hyperbola method

The region \(ab\le x\) is symmetric in \(a,b\). Let \(y=\sqrt x\). Counting points with \(a\le y\), then those with \(b\le y\), then subtracting the square counted twice,

Replace \(\lfloor x/a\rfloor\) by \(x/a+O(1)\) and use \(\sum_{a\le y}1/a=\log y+\gamma+O(1/y)\). The first sum is \(x\log x+2\gamma x+O(\sqrt x)\), and \(\lfloor y\rfloor^2=x+O(\sqrt x)\). Without the symmetry trick one gets only an error \(O(x)\). The exponent \(\tfrac12\) has been improved many times; the conjecture is that any exponent above \(\tfrac14\) works, and the problem remains open.

The Dirichlet series of \(d(n)\) is \(\sum d(n)n^{-s}=\zeta(s)^2\), and in general the product of two Dirichlet series is the series of the Dirichlet convolution \((f*g)(n)=\sum_{ab=n}f(a)g(b)\).

Italy and the Berlin school, 1843–1855

In 1843 Jacobi fell seriously ill with diabetes. Dirichlet persuaded the king of Prussia to fund a stay in Italy, and in the autumn he travelled south with Jacobi, Carl Wilhelm Borchardt, the geometer Jakob Steiner and the young Swiss mathematician Ludwig Schläfli, who served as interpreter and whom Dirichlet tutored on the way. The party reached Rome in November 1843. The Dirichlets stayed in Italy until the spring of 1845; their daughter Flora was born in Florence that February. In 1846 he declined a call to Heidelberg after Berlin improved his salary.

The teacher

Dirichlet was among the first German professors to lecture regularly on advanced research: number theory, definite integrals, potential theory and partial differential equations. His lectures were clear and complete, and students crossed Germany to hear them. Leopold Kronecker took his doctorate under him in 1845 and Rudolf Lipschitz in 1853. Gotthold Eisenstein and Ernst Kummer belonged to his Berlin circle. Bernhard Riemann studied in Berlin from 1847 to 1849 and heard Dirichlet, whose influence on him was decisive: Riemann’s 1854 habilitation thesis on trigonometric series begins from Dirichlet’s 1829 theorem.

The burden was heavy. Besides the university he still taught at the Military School, and in 1853 he complained to Kronecker of thirteen lectures a week. In July 1849 he travelled to Göttingen for the golden jubilee of Gauss’s doctorate, one of the few mathematicians present.

Göttingen, death and legacy, 1855–1859

Gauss died on 23 February 1855. Göttingen wanted Dirichlet as his successor, and when Berlin was slow to respond to his requests he accepted. He moved in the autumn of 1855; Ernst Kummer came from Breslau to take his Berlin chair. In Göttingen the Dirichlets bought a house with a garden, where Rebecka held gatherings with music in the Mendelssohn tradition. Riemann, then a young lecturer, and Richard Dedekind, who attended Dirichlet’s lectures for several years, became his close companions. Dedekind wrote that his almost daily contact with Dirichlet was the most useful part of his education. In 1855 Dirichlet was elected a foreign member of the Royal Society and received the order Pour le Mérite on Humboldt’s recommendation.

In the summer of 1858 he suffered a heart attack while at Montreux in Switzerland and returned to Göttingen with difficulty. While he lay ill, Rebecka died suddenly of a stroke on 1 December 1858. Dirichlet died on 5 May 1859, aged 54, and was buried in the Bartholomäus cemetery in Göttingen. Riemann succeeded him in the chair.

Legacy

• Analytic number theory: characters, L-functions and Dirichlet series are the language of the prime number theorem, the Riemann hypothesis and the Langlands programme.
• Algebraic number theory: the class number formula and the unit theorem are pillars of the subject. Dedekind’s edition of the Vorlesungen über Zahlentheorie (1863), with its supplements, introduced the theory of ideals.
• Analysis: the first rigorous convergence theorem for Fourier series, the general concept of function, the Dirichlet kernel and Dirichlet’s test for series.
• PDE and the calculus of variations: the Dirichlet problem, the Dirichlet energy and Dirichlet boundary conditions.
• Combinatorics and approximation: the pigeonhole principle and Dirichlet’s approximation theorem, the start of the geometry of numbers later developed by Hermann Minkowski.
• Honours: a lunar crater bears his name. His collected works were edited by Kronecker and Lazarus Fuchs (1889–1897).

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Johann Peter Gustav Lejeune Dirichlet, MacTutor History of Mathematics.
2. J. Elstrodt, The Life and Work of Gustav Lejeune Dirichlet (1805–1859), Clay Mathematics Proceedings 7 (2007).
3. O. Ore, Dirichlet, Gustav Peter Lejeune, Dictionary of Scientific Biography.
4. Peter Gustav Lejeune Dirichlet, Wikipedia.
5. Mathematics Genealogy Project entry.
6. U. C. Merzbach, Dirichlet: A Mathematical Biography, Birkhäuser, 2018.

Mathematics

1. P. G. L. Dirichlet, Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données, J. reine angew. Math. 4 (1829).
2. P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression … unendlich viele Primzahlen enthält, Abh. Akad. Berlin (1837). English translation by R. Stephan, arXiv:0808.1408.
3. P. G. L. Dirichlet, Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres, J. reine angew. Math. 19 (1839), 21 (1840).
4. P. G. L. Dirichlet, Zur Theorie der complexen Einheiten, Ber. Akad. Berlin (1846).
5. P. G. L. Dirichlet, Vorlesungen über Zahlentheorie, ed. R. Dedekind, 1863; English translation by J. Stillwell, AMS, 1999.
6. H. Davenport, Multiplicative Number Theory, 3rd ed., Springer, 2000.
7. J.-P. Serre, A Course in Arithmetic, Springer, 1973 (chapter VI on Dirichlet’s theorem).
8. A. F. Monna, Dirichlet’s Principle: A Mathematical Comedy of Errors and its Influence on the Development of Analysis, Oosthoek, 1975.