Analytic Number Theory Clay Mathematics Institute

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Clay Mathematics Proceedings,Analytic Number Theory. A Tribute to,Gauss and Dirichlet,William Duke,Yuri Tschinkel. American Mathematical Society,Clay Mathematics Institute. Foreword vii, The Life and Work of Gustav Lejeune Dirichlet 1805 1859 1. Ju rgen Elstrodt, An overview of Manin s conjecture for del Pezzo surfaces 39.
T D Browning, The density of integral solutions for pairs of diagonal cubic equations 57. Jo rg Bru dern and Trevor D Wooley,Second moments of GL2 automorphic L functions 77. Adrian Diaconu and Dorian Goldfeld,CM points and weight 3 2 modular forms 107. Jens Funke, The path to recent progress on small gaps between primes 129. D A Goldston J Pintz and C Y Y ld r m,Negative values of truncations to L 1 141.
Andrew Granville and K Soundararajan,Long arithmetic progressions of primes 149. Heegner points and non vanishing of Rankin Selberg L functions 169. Philippe Michel and Akshay Venkatesh, Singular moduli generating functions for modular curves and surfaces 185. Rational points of bounded height on threefolds 207. Per Salberger,Reciprocal Geodesics 217,Peter Sarnak. The fourth moment of Dirichlet L functions 239,K Soundararajan. The Gauss Class Number Problems 247, The year 2005 marked the 150th anniversary of the death of Gauss as well as.
the 200th anniversary of the birth of Dirichlet who became Gauss s successor at. Go ttingen In honor of these occasions a conference was held in Go ttingen from. June 20 to June 24 2005 These are the proceedings of this conference. In view of the enormous impact both Gauss and Dirichlet had on large areas of. mathematics anything even approaching a comprehensive representation of their. in uence in the form of a moderately sized conference seemed untenable Thus it. was decided to concentrate on one subject analytic number theory that could be. adequately represented and where their in uence was profound Indeed Dirichlet. is known as the father of analytic number theory The result was a broadly based. international gathering of leading number theorists who reported on recent advances. in both classical analytic number theory as well as in related parts of number theory. and algebraic geometry It is our hope that the legacy of Gauss and Dirichlet in. modern analytic number theory is apparent in these proceedings. We are grateful to the American Institute of Mathematics and the Clay Math. ematics Institute for their support,William Duke and Yuri Tschinkel. November 2006,Gustav Peter Lejeune Dirichlet, Courtesy of Nieders chsische Staats und Universit tsbibliothek G ttingen Sammlung Voit Lejeune Dirichlet Nr 2. Clay Mathematics Proceedings,Volume 7 2007, The Life and Work of Gustav Lejeune Dirichlet 1805 1859. Ju rgen Elstrodt, Dedicated to Jens Mennicke my friend over many years. Introduction 2,1 Family Background and School Education 2.
2 Study in Paris 4,3 Entering the Prussian Civil Service 7. 4 Habilitation and Professorship in Breslau 9,5 Transfer to Berlin and Marriage 12. 6 Teaching at the Military School 14, 7 Dirichlet as a Professor at the University of Berlin 15. 8 Mathematical Works 18,9 Friendship with Jacobi 28. 10 Friendship with Liouville 29,11 Vicissitudes of Life 30.
12 Dirichlet in Go ttingen 31,Conclusion 33,References 34. 2000 Mathematics Subject Classi cation Primary 01A55 Secondary 01A70. c 2007 Ju rgen Elstrodt,2 JU RGEN ELSTRODT,Introduction. The great advances of mathematics in Germany during the rst half of the nine. teenth century are to a predominantly large extent associated with the pioneering. work of C F Gau 1777 1855 C G J Jacobi 1804 1851 and G Lejeune Dirich. let 1805 1859 In fact virtually all leading German mathematicians of the second. half of the nineteenth century were their disciples or disciples of their disciples This. holds true to a special degree for Jacobi and Dirichlet who most successfully intro. duced a new level of teaching strongly oriented to their current research whereas. Gau had a real dislike of teaching at least at the poor level which was pre. dominant when Gau started his career The leading role of German mathematics. in the second half of the nineteenth century and even up to the fateful year 1933. would have been unthinkable without the foundations laid by Gau Jacobi and. Dirichlet But whereas Gau and Jacobi have been honoured by detailed biogra. phies e g Du Koe a similar account of Dirichlet s life and work is still a. desideratum repeatedly deplored In particular there exist in English only a few. mostly rather brief articles on Dirichlet some of which are unfortunately marred. by erroneous statements The present account is intended as a rst attempt to. remedy this situation,1 Family Background and School Education. Johann Peter Gustav Lejeune Dirichlet to give him his full name was born in. Du ren approximately halfway between Cologne and Aachen Aix la Chapelle. on February 13 1805 He was the seventh1 and last child of Johann Arnold Lejeune. Dirichlet 1762 1837 and his wife Anna Elisabeth ne e Lindner 1768 1868. Dirichlet s father was a postmaster merchant and city councillor in Du ren The. o cial name of his profession was commissaire de poste After 1807 the entire. region of the left bank of the Rhine was under French rule as a result of the wars. with revolutionary France and of the Napoleonic Wars Hence the members of the. Dirichlet family were French citizens at the time of Dirichlet s birth After the. nal defeat of Napole on Bonaparte at Waterloo and the ensuing reorganization of. Europe at the Congress of Vienna 1814 1815 a large region of the left bank of. the Rhine including Bonn Cologne Aachen and Du ren came under Prussian rule. and the Dirichlet family became Prussian citizens, Since the name Lejeune Dirichlet looks quite unusual for a German family we. brie y explain its origin2 Dirichlet s grandfather Antoine Lejeune Dirichlet 1711. 1784 was born in Verviers near Lie ge Belgium and settled in Du ren where he. got married to a daughter of a Du ren family It was his father who rst went. under the name Lejeune Dirichlet meaning the young Dirichlet in order to. di erentiate from his father who had the same rst name The name Dirichlet or. Derichelette means from Richelette after a little town in Belgium We mention. this since it has been purported erroneously that Dirichlet was a descendant of a. Hensel H 1 vol 1 p 349 says that Dirichlet s parents had 11 children Possibly this. number includes children which died in infancy, For many more details on Dirichlet s ancestors see BuJZ.
THE LIFE AND WORK OF GUSTAV LEJEUNE DIRICHLET 1805 1859 3. French Huguenot family This was not the case as the Dirichlet family was Roman. The spelling of the name Lejeune Dirichlet is not quite uniform Dirichlet himself. wrote his name Gustav Lejeune Dirichlet without a hyphen between the two parts. of his proper name The birth place of Dirichlet in Du ren Weierstra e 11 is marked. with a memorial tablet, Kummer Ku and Hensel H 1 vol 1 inform us that Dirichlet s parents gave their. highly gifted son a very careful upbringing This beyond doubt would not have been. an easy matter for them since they were by no means well o Dirichlet s school. and university education took place during a period of far reaching reorganization. of the Prussian educational system His school and university education however. show strong features of the pre reform era when formal prescriptions hardly existed. Dirichlet rst attended an elementary school and when this became insu cient a. private school There he also got instruction in Latin as a preparation for the sec. ondary school Gymnasium where the study of the ancient languages constituted. an essential part of the training Dirichlet s inclination for mathematics became. apparent very early He was not yet 12 years of age when he used his pocket money. to buy books on mathematics and when he was told that he could not understand. them he responded anyhow that he would read them until he understood them. At rst Dirichlet s parents wanted their son to become a merchant When he. uttered a strong dislike of this plan and said he wanted to study his parents gave. in and sent him to the Gymnasium in Bonn in 1817 There the 12 year old boy. was entrusted to the care and supervision of Peter Joseph Elvenich 1796 1886 a. brilliant student of ancient languages and philosophy who was acquainted with the. Dirichlet family Sc 1 Elvenich did not have much to supervise for Dirichlet. was a diligent and good pupil with pleasant manners who rapidly won the favour. of everybody who had something to do with him For this trait we have lifelong. numerous witnesses of renowned contemporaries such as A von Humboldt 1769. 1859 C F Gau C G J Jacobi Fanny Hensel ne e Mendelssohn Bartholdy 1805. 1847 Felix Mendelssohn Bartholdy 1809 1847 K A Varnhagen von Ense 1785. 1858 B Riemann 1826 1866 R Dedekind 1831 1916 Without neglecting his. other subjects Dirichlet showed a special interest in mathematics and history in. particular in the then recent history following the French Revolution It may be. assumed that Dirichlet s later free and liberal political views can be traced back to. these early studies and to his later stay in the house of General Foy in Paris see. After two years Dirichlet changed to the Jesuiter Gymnasium in Cologne Elvenich. became a philologist at the Gymnasium in Koblenz Later he was promoted to. professorships at the Universities of Bonn and Breslau and informed Dirichlet. during his stay in Bonn about the state of a airs with Dirichlet s doctor s diploma. In Cologne Dirichlet had mathematics lessons with Georg Simon Ohm 1789 1854. well known for his discovery of Ohm s Law 1826 after him the unit of electric. resistance got its name In 1843 Ohm discovered that pure tones are described by. purely sinusoidal oscillations This nding opened the way for the application of. Fourier analysis to acoustics Dirichlet made rapid progress in mathematics under. Ohm s guidance and by his diligent private study of mathematical treatises such. 4 JU RGEN ELSTRODT, that he acquired an unusually broad knowledge even at this early age He attended. the Gymnasium in Cologne for only one year starting in winter 1820 and then. left with a school leaving certi cate It has been asserted that Dirichlet passed. the Abitur examination but a check of the documents revealed that this was not. the case Sc 1 The regulations for the Abitur examination demanded that the. candidate must be able to carry on a conversation in Latin which was the lingua. franca of the learned world for centuries Since Dirichlet attended the Gymnasium. just for three years he most probably would have had problems in satisfying this. crucial condition Moreover he did not need the Abitur to study mathematics. which is what he aspired to Nevertheless his lacking the ability to speak Latin. caused him much trouble during his career as we will see later In any case Dirichlet. left the Gymnasium at the unusually early age of 16 years with a school leaving. certi cate but without an Abitur examination, His parents now wanted him to study law in order to secure a good living to their. son Dirichlet declared his willingness to devote himself to this bread and butter. education during daytime but then he would study mathematics at night After. this his parents were wise enough to give in and gave their son their permission to. study mathematics,2 Study in Paris, Around 1820 the conditions to study mathematics in Germany were fairly bad. for students really deeply interested in the subject Lo The only world famous. mathematician was C F Gau in Go ttingen but he held a chair for astronomy. and was rst and foremost Director of the Sternwarte and almost all his courses. were devoted to astronomy geodesy and applied mathematics see the list in Du. p 405 Moreover Gau did not like teaching at least not on the low level. which was customary at that time On the contrary the conditions in France. were in nitely better Eminent scientists such as P S Laplace 1749 1827 A M. Legendre 1752 1833 J Fourier 1768 1830 S D Poisson 1781 1840 A L. Cauchy 1789 1857 were active in Paris making the capital of France the world. capital of mathematics Hensel H 1 vol 1 p 351 informs us that Dirichlet s. parents still had friendly relations with some families in Paris since the time of the. French rule and they let their son go to Paris in May 1822 to study mathematics. Dirichlet studied at the Colle ge de France and at the Faculte des Sciences where. he attended lectures of noted professors such as S F Lacroix 1765 1843 J B. Biot 1774 1862 J N P Hachette 1769 1834 and L B Franc ur 1773 1849. He also asked for permission to attend lectures as a guest student at the famous. E cole Polytechnique But the Prussian charge d a aires in Paris refused to ask for. such a permission without the special authorization from the Prussian minister of. religious educational and medical a airs Karl Freiherr von Stein zum Altenstein. 1770 1840 The 17 year old student Dirichlet from a little provincial Rhenisch. town had no chance to procure such an authorization. More details about Dirichlet s courses are apparently not known We do know that. Dirichlet besides his courses devoted himself to a deep private study of Gau. masterpiece Disquisitiones arithmeticae At Dirichlet s request his mother had pro. modern analytic number theory is apparent in these proceedings We are grateful to the American Institute of Mathematics and the Clay Math ematics Institute for their support William Duke and Yuri Tschinkel November 2006 vii Courtesy of Nieders chsische Staats und Universit tsbibliothek G ttingen Sammlung Voit Lejeune Dirichlet Nr 2 Gustav Peter Lejeune Dirichlet Clay Mathematics

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