Analytic Number Theory - Mathematical Institute - University of Oxford

266 Pages·2008·2.63 MB·English

Clay Mathematics Proceedings
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4-color process 264 pages on 50 lb stock • 1/2 inch spine
Analytic Number Theory
A Tribute to
Gauss and Dirichlet

Clay Mathematics Proceedings
Volume 7
Analytic Number Theory
A Tribute to
Gauss and Dirichlet
William Duke
Yuri Tschinkel
Editors
American Mathematical Society
Clay Mathematics Institute

Contents
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 GL automorphic L-functions 77
2
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

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 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.
InviewoftheenormousimpactbothGaussandDirichlethadonlargeareasof
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
internationalgatheringofleadingnumbertheoristswhoreportedonrecentadvances
inbothclassicalanalyticnumbertheoryaswellasinrelatedpartsofnumbertheory
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
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Gustav-Peter Lejeune Dirichlet
ClayMathematicsProceedings
Volume7,2007
The Life and Work of Gustav Lejeune Dirichlet (1805–1859)
Ju¨rgen 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 Go¨ttingen 31
Conclusion 33
References 34
2000 Mathematics SubjectClassification. Primary01A55,Secondary01A70.
(cid:1)c 2007Ju¨rgenElstrodt
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