Clay Mathematics Proceedings Volume 7 American Mathematical Society Clay Mathematics Institute Analytic Number Theory A Tribute to Gauss and Dirichlet 7 AMS CMI Duke and Tschinkel, Editors 264 pages on 50 lb stock • 1/2 inch spine Analytic Number Theory A Tribute to Gauss and Dirichlet William Duke Yuri Tschinkel Editors CMIP/7 www.ams.org www.claymath.org 4-color process Articles in this volume are based on talks given at the Gauss– Dirichlet Conference held in Göttingen on June 20–24, 2005. The conference commemorated the 150th anniversary of the death of C F. Gauss and the 200th anniversary of the birth of J L. Dirichlet. The volume begins with a definitive summary of the life and work of Dirichlet and continues with thirteen papers by leading experts on research topics of current interest in number theory that were directly influenced by Gauss and Dirichlet. Among the topics are the distribution of primes (long arithmetic progres- sions of primes and small gaps between primes), class groups of binary quadratic forms, various aspects of the theory of L-func- tions, the theory of modular forms, and the study of rational and integral solutions to polynomial equations in several variables. Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Analytic Number Theory A Tribute to Gauss and Dirichlet Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com American Mathematical Society Clay Mathematics Institute Clay Mathematics Proceedings Volume 7 Analytic Number Theory A Tribute to Gauss and Dirichlet William Duke Yuri Tschinkel Editors Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Contents Foreword vii The Life and Work of Gustav Lejeune Dirichlet (1805–1859) 1 J ¨ urgen 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 J ¨ org Br ¨ udern and Trevor D. Wooley Second moments of GL 2 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. Yıldırım Negative values of truncations to L(1,χ) 141 Andrew Granville and K. Soundararajan Long arithmetic progressions of primes 149 Ben Green 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 Ken Ono 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 H. M. Stark v Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Foreword The year 2005 marked the 150th anniversary of the death of Gauss as well as the 200th anniversary of the birth of Dirichlet, who b ecame Gauss’s successor at G¨ottingen. In honor of these occasions, a conference was held in G¨ottingen 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 influence 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 influence 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 vii Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Courtesy of Niedersächsische Staats - und Universitätsbibliothek Göttingen, Sammlung Voit: Lejeune-Dirichlet, Nr. 2. Gustav-Peter Lejeune Dirichlet Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Clay Mathematics Proceedings Volume 7, 2007 The Life and Work of Gustav Lejeune Dirichlet (1805–1859) J¨urgen Elstrodt Dedicated to Jens Mennicke, my friend over many years Contents 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 G¨ottingen 31 Conclusion 33 References 34 2000 Mathematics Subject Classification. Primary 01A55, Secondary 01A70. c  2007 J¨urgen Elstrodt 1 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com [...]... offered Dirichlet a teaching position at the University of Breslau (Silesia, now Wroclaw, Poland) with an opportunity for a Habilitation (qualification examination for lecturing at a university) and a modest annual salary of 400 talers, which was the usual starting salary of an associate professor at that time (This was not too bad an offer for a 21-year-old young man without any final examination.) Von Altenstein... grey (perhaps 45 years old), with a somewhat harsh voice and rather deaf: it was early, he was unwashed, and unshaved (what of him required shaving), with his ‘Schlafrock’, slippers, cup of coffee and cigar I thought, as we sat each at an end of the sofa, and the smoke of our cigars carried question and answer to and fro, and intermingled in graceful curves before it rose to the ceiling and mixed with... hands, mostly shut He uses no notes, inside his hands he sees an imaginary calculation, and reads it out to us — that we understand it as well as if we too saw it I like that kind of lecturing.” — After Hirst called on Dirichlet and was “met with a very hearty reception”, he noted in his diary on October 13, 1852 ([GW], p 622): “He is a rather tall, lankylooking man, with moustache and beard about to. .. wanted Dirichlet to move to Breslau just a few weeks later since there was a vacancy He added, if Dirichlet had not yet passed the doctoral examination, he might send an application to the philosophical faculty of the University of Bonn which would grant him all facilities consistent with the rules ([Sc.1]) The awarding of the doctorate, however, took more time than von Altenstein and Dirichlet had... many mathematical matters completely agreed with Fourier’s, notably on the foundations of geometry For Dirichlet, the first task in Breslau was to habilitate (qualify as a university lecturer) According to the rules in force he had a) to give a trial lecture, b) to write a thesis (Habilitationsschrift) in Latin, and c) to defend his thesis in a public disputation to be held in Latin Conditions a) and. .. French capital, and this enabled Dirichlet to gain self-assurance in his social bearing, which was of notable importance for his further life Dirichlet soon became acquainted with his academic teachers His first work of academic character was a French translation of a paper by J .A Eytelwein (1764– 1848), member of the Royal Academy of Sciences in Berlin, on hydrodynamics ([Ey]) Dirichlet s teacher Hachette... manner Likewise, from a letter of Gauß to Olbers ([O.2], p 479), we know that Gauß too was very much pleased to meet Dirichlet in person, and he expresses his great satisfaction that his recommendation had apparently helped Dirichlet to acquire his appointment Gauß also tells something about the topics of the conversation, and he says that he was surprised to learn from Dirichlet, that his (i.e., Gauß’)... 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 certificate but without an Abitur examination His parents now wanted him to study law in order to secure a good living to their... world-famous mathematician was C.F Gauß in G¨ttingen, but he held a chair for astronomy o and was first 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 ff.) Moreover, Gauß did not like teaching – at least not on the low level which was customary at that time On the contrary, the conditions in France... research expedition with A Bonpland (1773–1858) to South and Central America This expedition had earned him enormous world-wide fame, and he became a corresponding member of the French Academy in 1804 and a foreign member in 1810 Von Humboldt took an exceedingly broad interest in the natural sciences and beyond that, and he made generous good use of his fame to support young talents in any kind of art . Clay Mathematics Proceedings Volume 7 American Mathematical Society Clay Mathematics Institute Analytic Number Theory A Tribute to Gauss and Dirichlet 7 AMS CMI Duke and Tschinkel, Editors 264. Editors 264 pages on 50 lb stock • 1/2 inch spine Analytic Number Theory A Tribute to Gauss and Dirichlet William Duke Yuri Tschinkel Editors CMIP/7 www.ams.org www.claymath.org 4-color process Articles. (qualification examination for lecturing at a university) and a modest annual salary of 400 talers, which was the usual starting salary of an associate professor at that time. (This was not too bad an offer