Allen Shields*

Lejeune Dirichlet and the Birth of Analytic Number Theory: 1837-1839

It is approximately 150 years since Dirichlet's funda- mental paper [18371] appeared, in which he proved that any ar i thmet ic p rogress ion {a + nq}, n = 0,1,2 . . . . . for which a and q are relatively prime, contains infinitely many prime numbers . He pre- sented his result to the Royal Prussian Academy of

Sc iences in Berlin on 27 July 1837, and it was pub- lished in the Abhandlungen of the Academy for that year. Howeve r , the Abhandlung was not actually printed until 1839, at which point it became available to a wider public. Also, as Davenport [1980] points out (page 1), the proof in this first paper of Dirichlet was complete only in case q was a prime. For the general case, Dirichlet had to assume his class number for- mula (for the number of inequivalent quadratic forms in two variables over the integers, with fixed discrimi- nant). He publ ished the proof of this formula in [1839-40]. The theorem on the infinitude of primes in arithmetic progressions had been conjectured earlier by Legendre, and used by him as an unproved lemma in some of his researches (see Dirichlet [18371], Werke, p. 316).

Davenport begins his book with the words: "Ana- lytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet's memoir of 1837 on the existence of primes in a given arithmetic progression."

Dirichlet based his proof in part on a proof by Euler of the infinitude of primes; Euler's method proved the stronger result that Y~ p- ~ = ~, where the summation is over all prime numbers (see Euler [1748], Chap. 15). Dirichlet proved the corresponding result that the sum of the reciprocals of the prime numbers in an arith- metic progression is infinite. (See Dirichlet [18372], 309-310.)

* Column editor's address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003 USA

Euler argued as follows. First, if p is a prime number then (1 - p-1)-1 = y~ p - , (0 ~ n < o0). Let M be a given (large) positive number, and choose N so that Y~ n-1 > M, where the summation is for 1 ~ n ~ N. Form the product 11 (1 - p-1)-1 over all primes p ~ N. If we multiply the corresponding infinite series, then we may rearrange their product in any order because of the absolute convergence. In particular, we see that the product of the series contains all reciprocal in- tegers n-~, n ~ N. Thus the product is greater than M. Taking the rec iproca l we see that the p r o d u c t II (1 - p- l ) over all primes is zero. This is equivalent (by a standard relation between products and series) to the divergence of ~ p-1.

These ideas lead to Euler's formula: II (1 - p-S)-1 = E n - S , s > 1.

Dirichlet had the idea of introducing what today are called group characters and using them to obtain anal- ogous formulae. If q is a prime number, then the in- tegers 1,2 . . . . . q - 1 form a group under multiplica- tion mod q. Let X be a character of this group. We de- fine • for all positive integers n by putting x(n) = 0 if n is a multiple of q, and x(n) = x(j) if n and j are con- gruent mod q. Dirichlet formed the series Lx(S ) =

x(n) n-s, s > 1 (summed over all positive integers n) and obtained the analogue of Euler's formula:

L• = 1-I (1 - •

where the product is over all primes p (p # q). Dirichlet used the letter L for these functions, and it has been used ever since.

The reader wishing more precise details should con- sult Chapter I of Davenport [1980]. After some prelim- inary simplification the crucial point in the proof consists in showing that, if • is not the identity char- acter (that is, the character such that • = 1, n not a multiple of q), then Lx(s ) remains bounded as s ap- proaches 1.

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 �9 1989 Springer-Verlag New York 7

Gustav Peter

Lejeune Dirichlet

We mention that for certain special arithmetic pro- gressions (for example, {4n + 3}), a simple proof can be given, similar to Euclid's proof that there are infi- nitely many primes. See Chapter 2, Section 3, of Hardy and Wright [1938, 1960].

In van der Waerden ' s History of Algebra [1985], Chapter 12, he says that the theory of group characters begins with Gauss 's Disquisitiones Arithmeticae, Sec- tions 228-233. Dirichlet (who studied Gauss's book very closely, see below) carried the theory further, especially in the first paper cited in Dirichlet [1839- 40]. Dedekind (who was Dirichlet's student) gave an exposition of the results of this paper in the fourth supplement to Dirichlet's Vorlesungen iiber Zahlen- theorie, 3rd edition, 1879. In the famous tenth supple- ment Dedekind carries the theory further. In a letter to Frobenius he wrote: "After all this [i.e., after Di- richiet's investigations], it was not much to introduce the concept and name of characters for every Abelian group, as I did in the third edition of Dirichlet's Zah- lentheorie."

We turn now to some biographical information about Dirichlet. Much of this is taken from Biermann [1959], Kummer [1860], Rowe [1988], and Klein [1926]. The reader would do well to commence with Rowe.

Dirichlet's grandfather came from Verviers (Bel- gium) and emigrated to D/iren, between Aachen and K61n, in Germany (about 50 kilometers from Verviers), becoming a B/irger of D/iren in 1753. He made cloth. The name Dirichlet comes from "de Richelet" (Bier- mann, p. 8). Johann Peter Gustav Lejeune Dirichlet was born 13 February 1805 in DLiren; his father was the postmaster (Postkommissar) for the town.

In May 1822 he went to Paris to study mathematics; Paris was then the world center for mathematics. Le- gendre (1752-1833) was still active, and Fourier (1768-1830) and Poisson (1781-1840) were leading members of the Academy. One year later he was in- vited by General Foy, the leader of the Opposition in the Chamber of Deputies, to teach mathematics to his children. In addition to attending lectures, Dirichlet spent much time studying Gauss's Disquisitiones Arith- meticae. Kummer (p. 315-316) states that Dirichlet read it not once, or even several times, but throughout his life he reread it. It was never in his bookcase, but always on the table where he worked. More than twenty years after this extraordinary book had ap- peared (in 1801) no one had yet understood it. Even Legendre had to confess in the second edition of his Th~orie des nombres that he would have liked to have enriched his book with Gauss's results; however, the methods of this author were so peculiar that this was impossible without great digressions, or merely as- suming the role of a translator. Dirichlet was the first to understand this work thoroughly, but beyond that he found more natural proofs, thereby making the re- sults accessible to others and enabling him to find deep new results.

Dirichlet met Alexander yon Humboldt (1769-1859) in Paris in 1825 and told him that he would like to return to Germany, if there were a position for him. Von Humbold t had heard of Dirichlet's talent from Fourier. Fourier liked to gather talented young men around himself, and Dirichlet had been a part of this circle. Gauss (1777-1855) wrote a strong letter of rec- ommendation for Dirichlet. Von Humboldt wrote the Prussian Minister of Education, who replied that Di- richlet could go to the University of Breslau for his Ha- bilitation with an annual salary of four hundred talers as a Privatdozent. In addition he would receive sev- enty-five talers to cover the expense of moving to Breslau. Von Humboldt was not satisfied and wrote again suggesting that Dirichlet be given the title of professor , with a salary of six to seven h u n d r e d talers. Nothing came of this, however . Before ap- pointing Dirichlet (who was now twenty-one years old) as a Privatdozent the Minister felt compelled to check his background. It was not a liberal period and it would have been dangerous to appoint an unknown young man merely on the recommendation of yon Humboldt , who was himself considered somewhat liberal. The Minister of Education wrote to the Min- ister of the Interior asking if the police had anything against the appointment. They were suspicious of Di- richlet's connection to General Foy and inquiries were made in Paris. But in the end they decided there was nothing against him, and his connection with Foy had been for scientific rather than political purposes. He took up his duties in Breslau in April 1827.

In Breslau a curious problem awaited him. The fac-

8 THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

ulty rules for the Habilitation required a dissertation written in Latin, which he could do, together with an oral presentation and defense, also in Latin, which he could not. However, at Dirichlet's request the Minister excused him from the oral defense. Some of the fac- ulty were angered at this intervention from above and wrote the Minister (see Biermann, pp, 21-30).

On 1 April 1828 he was given the title of ausseror- dentlicher Professor in Breslau with the same salary of four hundred talers. He wished to go to Berlin, which was the center of mathematical life in Prussia. A posi- tion opened up at the Royal Military School in Berlin, and he was given leave from Breslau to teach there for a year. Faculty from the University of Berlin some- times taught at the Military School to supplement their income. Dirichlet's position had been held by Pro- fessor Ohm (1787-1854) (of "Ohm's Law" in elec- tricity), who had been teaching six hours a week for nine months for an annual salary of six hundred talers. Now the teaching load was increased to eight hours a week with no increase in salary. Professor Ohm announced that he would not teach eight hours un less he was paid eight h u n d r e d talers. They couldn't do this so he left, thereby opening the posi-

-tion for Dirichlet. In Easter 1829, he became a Privatdozent at the Uni-

versity of Berlin. In July 1831, he was promoted to aus- serordentlicher Professor, with a salary of four hundred talers; in addition, he continued teaching at the Mili- tary School.

In May 1832 he married Rebekka Mendelssohn- Bartholdy, the sister of the composer Felix. They had three sons and a daughter. Through the Mendels- sohns he was connected to a number of other mathe- maticians. For example , K u m m e r ' s wife Ottil ie (maiden name Mendelssohn-Bartholdy) was a cousin of Rebekka, and later H. A. Schwarz (1843-1921) mar- ried Kummer's daughter. Finally, Kurt Hensel (1861- 1941) was a great nephew of Rebekka.

'As far as I can tell from Biermann [1959] (see his list of names at the end of the article), Dedekind (1831- 1916) and Kronecker (1823-1891) were students of Dirichlet, and Eisenstein (1823-1852), Kummer (1810- 1893), and Riemann (1826-1866) were influenced by his lectures. Riemann came over from G6ttingen for a year to attend lectures in Berlin. Biermann (pp. 34-39) lists the courses Dirichlet taught at Berlin with the number of students in each course. For example, in the summer semester of 1829 he taught "The theory of series, as an introduction to higher analysis" to five students. He also offered a course called "Indeter- minate analysis" but only two students showed up and the course did not run. He seems to have offered two courses per semester (with two semesters per year), but if only two or three students reported, the course would not run. Mostly there were fewer than 20 students.

We digress to say a few words about A. L. Crelle (1780-1855), who had wide interests in technology as well as mathematics. Klein [1926; p. 94-96] states that Crelle never gave up mathematics despite his other wide-ranging interests, but that his mathematical work was without real significance. It had an encyclo- pedic character, which was really a tradition of the previous century but which was still widespread in Germany in that his work touched on many different topics in mathematics without going deeply into any- thing. He was a member of the Prussian Academy of Sciences. However, his real talents and contributions lay elsewhere. First of all he had a good eye for mathe- matical talent, and he labored to secure a position for Abel in Berlin (when the offer was finally sent in 1829 Abel had just died in Christiana). He also wrote the Prussian Minister in 1830 supporting Dirichlet for a permanent position in Berlin. Finally in 1839 he wrote supporting Dirichlet's promotion to Professor (Ordi- narius). But Crelle is especial ly r emembered for founding the Journal fiir die reine und angewandte Mathe- matik, still sometimes called Crelle's Journal, and editing the first fifty volumes. At the beginning there were serious financial difficulties, which he met partly from his own resources. After much effort he secured a state subsidy. The first volume contained five papers by Abel, one by Jacobi, and several by Steiner. In volume 3 (1828) there are papers by Dirichlet, M6bius, and Pl~icker.

Gustav Peter Lejeune Dirichlet

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989 9

In 1832 Dirichlet was elected a m e m b e r of the Academy of Sciences and became the first top-rank mathematician in the Academy since Lagrange left in 1787 to go to Paris. In 1834 his University salary was increased, in view of his "laudable effectiveness" (bei- fallswerthen Wirksamkeit), to six hundred talers. He still taught at the Military School for an additional six hundred talers. In 1839 he was promoted to Professor at the University, still with the same salary of six

The Minister replied that in v iew of Di- richlet's scientific achievements and his ac- tivity as a teacher for many years the min- istry had the sincere wish to grant him a suit- able increase in salary, but that unfortunately this was not possible at the present time.

hundred talers. Finally in 1842 this was increased to eight hundred talers. He still taught at the Military School, though the constant routine lectures there were burdensome. Apparent ly the salary of eight hundred talers at the University was below the official scale for a Professor, as Dirichlet wrote the Minister of Education in August 1846 requesting that he be paid the official salary. He says he will not comment on his research, but that he feels his teaching has been suc- cessful. When he came to Berlin there were no lectures in many of the most important areas of mathematics. Dirichlet worked to change all this, and it cost a great deal of effort but he feels he was successful. The Min- ister replied that in v iew of Dirichlet 's scientific achievements and his activity as a teacher for many years the ministry had the sincere wish to grant him a suitable increase in salary, but that unfortunately this was not possible at the present time.

Just at this point a new element was introduced into this situation. At the end of 1846 the government of Baden wrote to Dirichlet asking if he would consider receiving a call to a position in Heidelberg University. The faculty at Berlin instructed the rector to write the Minister, urging him to raise Dirichlet's salary, for otherwise they might lose him to a foreign university and his loss could not be replaced. Jacobi wrote to the Minister and to the King urging that Dirichlet be kept. In the letter to the King Jacobi said that it would be impossible to replace Dirichlet since, aside from Gauss in G6ttingen and Cauchy in Paris, his equal was not to be found. Dirichlet's loss and Bessel's recent death would mean that Prussia would no longer be so com- petitive in the exact sciences.

Jacobi also wrote von Humboldt giving him argu- ments to use with the Minister. We quote one passage:

He alone, not I, not Cauchy, not Gauss knew what a com- pletely rigorous mathematical proof is, indeed we first

learned this from him. When Gauss says he has proved something, it is very probable . . . . when Cauchy says it, you can bet equally well pro or contra, but when Dirichlet says it, it is certain. I prefer to leave myself out of this Deli- katessen.

In the end these efforts were successful: Dirichlet's salary at the University was raised from 800 talers to 1500 talers and he stayed in Berlin.

Biermann [1959; p. 69] notes that in March 1848 Di- richlet stood guard before the Palace of Prince William of Prussia as a member of the citizen's militia. The palace had been declared to be "national property." The Encyclopaedia Brittanica (11th edition, 1911, vol. 11, p. 866), in the article on Germany describes this pe- riod. In February 1847 Friedrich William IV of Prussia summoned a united diet of Prussia to meet. But as Metternich (the Austrian premier) predicted, this only served as a forum for demands for a constitution, and deadlock ensued. In February 1848 revolution broke out in Paris; in May 1848 Metternich fell from power. A riot broke out in Berlin, which was suppressed on March 15 by troops with but little bloodshed. Ac- cording to the encyclopedia article the king had an "'emotional and kindly temperament" and "shrank with horror from the thought of fighting his 'beloved Berliners'." When on the night of the eighteenth of March the fighting was renewed he entered into nego- tiations with the insurgents, which led to the with- drawal of the troops.

The next day Friedrich Wilhelm, with characteristic his- trionic versatility, was heading a procession round the streets of Berlin, wrapped in the German tricolour, and extolling in a letter to the indignant Tsar the consumma- tion of 'the glorious German revolution'.

Jacobi said that it would be impossible to re- place Dirichlet since, aside from Gauss in G6ttingen and Cauchy in Paris, his equal was n o t to be found.

Dirichlet found the teaching at the Military College too burdensome and asked to be excused from it, but this was not done. Then after Gauss's death in 1855 the government of Hanover wrote him asking if he would consider coming to G6tfingen as Gauss's suc- cessor. He replied that he would, unless Prussia ex- cused him from teaching at the Military School. The Prussian Minister of Education, however, decided to wait for an official offer from Hanover before re- sponding. When the offer came, he excused Dirichlet from further duties in the Military School, and raised his salary. But it was too late--Dirichlet felt bound by what he had wrriten to Hanover, and he left. Kummer was called as his successor in Berlin with a salary of fifteen hundred talers.

10 THE MATHEMATICAL INTELLIGENCER VOL. II, NO. 4, 1989

Commenting that the to ta l number of Di- richlet's publications was not so large, Gauss wrote that the works of Dirichlet are jewels, and that jewels are not weighed on a grocery scale.

Dirichlet taught at GOttingen until 1858 w h e n he fell ill on a trip to Switzerland. Returning to G6t t ingen he died of severe hear t sickness on 5 May 1859.

C o m m e n t i n g that the total n u m b e r of Dirichlet 's publications was not so large, Gauss wrote that the works of Dirichlet are jewels, and that jewels are not weighed on a grocery scale (Biermann, p. 5).

Bibliography

MR = Mathematical Reviews; JFM = Jahrbuch iiber die Fortschritte der Mathematik; ZBL = Zentralblatt fiir Math- ematik.

1. K.-R. Biermann [1959], Johann Peter Gustav Lejeune Dirichlet, Dokumente fiir sein Leben und Wirken (zum 100. Todestag), Abhandlungen der Deutsch. Akad. der Wissen. zu Berlin, Klasse ffir Math., Physik und Technik, Jahrgang 1959, Nr. 2, Akademie Verlag, Berlin.

2. H. Davenport [1980], Multiplicative number theory (2nd

ed.), Graduate Texts in Mathematics 74, Springer- Verlag, New York. (1st ed. 1967, Markham Publishers, Chicago.) MR 82m:10001

3. P. G. L. Dirichlet [18371], Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftli- chen Factor sind, unendlich viele Primzahlen enth/ilt. Abhandlungen der KfJniglich Preussischen Akademie der Wis- senschaflen zu Berlin, 45-81; Werke I, 313-342.

4. - - [18372], Beweis eines Satzes fiber die aritheme- tische Progression, Werke I, 312.

5. - - [1839-40], Recherches sur diverses applications de l'analyse infinit6simale a la th6orie des nombres, J. reine angew. Math. 19, 324-369; 21, 1-12 and 134-155; Werke I, 411-496.

6. - - [1889], Werke I, ed. L. Kronecker; II (1897), ed. L. Fuchs, Verlag Georg Reimer, Berlin. JFM 21, 16-17.

7. L. Euler [1848], Introductio in analysin infinitorum, vol. I, Bousquet, Lausanne.

8. G. H. Hardy and E. M. Wright, An Introduction to the theory of numbers, Oxford, University Press, London; 1st ed. 1938, 4th ed. 1960.

9. F. Klein [1926], Vorlesungen fiber die Entwicklung der Math- ematik im 19. Jahrhundert, Teil I, ed. R. Courant, O. Neu- gebauer, Julius Springer Verlag, Berlin. JFM 52, 22-24.

10. E. E. Kummer [1860], Ged/ichtnisrede auf Gustav Peter Lejeune Dirichlet, Abh. K6nig. Akad. Wissen. zu Berlin. Reprinted in Dirichlet's Werke, 309-344.

11. D. E. Rowe [1988], Gauss, Dirichlet, and the law of bi- quadratic reciprocity, Math. Intellig. 10, no. 2, 13-25.

12. B. L. van der Waerden [1985], A history of algebra, Springer-Verlag, Berlin, Heidelberg. MR 87e:01001.

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