Aubert / Bombieri / Goldfeld Number Theory, Trace Formulas and Discrete Groups

Symposium in Honor of Atle Selberg, Oslo, Norway, July 14-21, 1987
1. Auflage 2014
ISBN: 978-1-4832-1623-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 532 Seiten, Web PDF

Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg Oslo, Norway, July 14-21, 1987 is a collection of papers presented at the 1987 Selberg Symposium, held at the University of Oslo. This symposium contains 30 lectures that cover the significant contribution of Atle Selberg in the field of mathematics. This book is organized into three parts encompassing 29 chapters. The first part presents a brief introduction to the history and developments of the zeta-function. The second part contains lectures on Selberg's considerable research studies on understanding the principles of several aspects of mathematics, including in modular forms, the Riemann zeta function, analytic number theory, sieve methods, discrete groups, and trace formula. The third part is devoted to Selberg's further research works on these topics, with particular emphasis on their practical applications. Some of these research studies, including the integral representations of Einstein series and L-functions; first eigenvalue for congruence groups; the zeta function of a Kleinian group; and the Waring's problem are discussed. This book will prove useful to mathematicians, researchers, and students.

Autoren/Hrsg.

Aubert, Karl Egil

Bombieri, Enrico

Goldfeld, Dorian

Inhaltsverzeichnis

1;Front Cover;1
2;Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg
Oslo, Norway, July 14—21, 1987;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;10
6;Participants in the Selberg Symposium, 1987;14
7;Preface;18
8;Foreword;20
9;PART I: HISTORICAL INTRODUCTION;22
9.1;Chapter 1.
Prehistory of the Zeta-Function;24
9.1.1;References;32
10;PART II: SURVEY LECTURES ON SELBERG'S WORK;34
10.1;Chapter 2.
The Trace Formula and Hecke Operators;36
10.1.1;1;36
10.1.2;2;40
10.1.3;3;44
10.1.4;4;48
10.1.5;References;51
10.1.6;Note Added in Proof;52
10.2;Chapter 3.
Selberg's Sieve and Its Applications;54
10.2.1;1;54
10.2.2;2. Selberg's Sieve;61
10.2.3;3. Further Developments;68
10.2.4;References;72
10.3;Chapter 4. The Rankin—Selberg Method: A Survey;74
10.3.1;1. The Rankin–Selberg Method for SL(2, Z);74
10.3.2;2. The Rankin–Selberg Method for the General Linear Groups;91
10.3.3;3. The Classical Groups;110
10.3.4;4. The Metaplectic Group;126
10.3.5;Acknowledgements;128
10.3.6;References;129
10.3.7;Added in Proof;134
10.4;Chapter 5.
On the Base Change Problem: After J. Arthur and L. Clozel;136
10.4.1;1. Automorphic L-functions;136
10.4.2;2. Induction and Base Change;138
10.4.3;3. The Distribution I(f);140
10.4.4;4. The Distribution I(f',t);142
10.4.5;5. Equality of the Distributions;143
10.4.6;6. Exploiting the Basic Identity;145
10.4.7;7. Concluding Remarks;148
10.4.8;References;149
10.5;Chapter 6.
Eisenstein Series, the Trace Formula, and the Modern Theory of Automorphic Forms;150
10.5.1;1. Eisenstein Series and Automorphic L-functions;150
10.5.2;2. The Structure of Trace Formulas and their Comparison;159
10.5.3;References;179
10.6;Chapter 7.
Selberg's Work on the Zeta-Function;182
10.6.1;1. Zeros on or Near the Critical Line;182
10.6.2;2. The Distribution of log .(1/2 + it);188
10.6.3;3. Other Work;191
10.6.4;References;192
10.7;Chapter 8.
Selberg's Work on the Arithmeticity of Lattices and its Ramifications;194
10.7.1;1. Introduction;194
10.7.2;2. Selberg's Rigidity Theorem;196
10.7.3;3. Strong Rigidity;197
10.7.4;4. Margulis' Arithmeticity Theorem;200
10.7.5;5. Existence of Nonarithmetic Lattices in O(ra, 1) for n > 1;202
10.7.6;6. Nonarithmetic Lattices in U(n, 1);202
10.7.7;7. Other Rigidity Phenomena;204
10.7.8;References;205
11;PART III: RESEARCH ANNOUNCEMENTS;210
11.1;Chapter 9.
Mean Values of the Riemann Zeta-Function with Application to the Distribution of Zeros;212
11.1.1;1;212
11.1.2;2;213
11.1.3;3;215
11.1.4;4;219
11.1.5;5. Concluding Remarks;224
11.1.6;References;225
11.1.7;Note added in proof;226
11.2;Chapter 10. Geometric Ramanujan Conjecture and Drinfeld Reciprocity Law;228
11.2.1;Integrality Conjecture;230
11.2.2;Purity Theorem;231
11.2.3;Reciprocity Law;232
11.2.4;Existence Theorem 1;233
11.2.5;Converse Theorem;234
11.2.6;Drinfeld Reciprocity Law;237
11.2.7;Grothendieck Fixed Point Formula;238
11.2.8;Lefschetz Fixed-Point Formula;239
11.2.9;Deligne's Conjecture;239
11.2.10;Trace Formula;241
11.2.11;Congruence Relations;243
11.2.12;Corollary;244
11.2.13;References;244
11.3;Chapter 11. On the Brun—Titchmarsh Theorem;246
11.3.1;Results on Average;250
11.3.2;References;254
11.3.3;Note Added in Proof;255
11.4;Chapter 12. A Double Sum Over Primes and Zeros of the Zeta-Function;256
11.4.1;1. Application of the Landau–Gonek Formula and Reduction to BR,S;258
11.4.2;2. Application of the Second Moment Bounds;259
11.4.3;3. Application of Guinand's Formula and MC;261
11.4.4;References;266
11.5;Chapter 13. Integral Representations of Eisenstein series and L-functions;268
11.5.1;1. Automorphic Forms on Some Classical Groups;271
11.5.2;2. Statement of the Theorems;279
11.5.3;3. Proof of the Main Formula;284
11.5.4;References;291
11.6;Chapter 14. Recent Results on Automorphic L-Functions;292
11.6.1;Introduction;292
11.6.2;1. The Method of Zeta-Integrals;293
11.6.3;2. The Method of Eisenstein Series and Local Coefficients;295
11.6.4;3. Comparison of the Methods;297
11.6.5;4. Combining the Methods;300
11.6.6;References;305
11.7;Chapter 15.
Explicit Formulae as Trace Formulae;308
11.7.1;1. Introduction;308
11.7.2;2. Notation;309
11.7.3;3. The Explicit Formula;311
11.7.4;References;315
11.8;Chapter 16.
Some Remarks on the Sieve Method;316
11.8.1;1. Introduction;316
11.8.2;2. The Linear Sieve;319
11.8.3;3. The Weighted Linear Sieve;324
11.8.4;4. A Weighted Sieve Following Rosser;327
11.8.5;5. Weight-warping;329
11.8.6;6. Selberg's .2 Method;332
11.9;Chapter 17.
Critical Zeros of GL(2) L-Functions;336
11.9.1;1. Introduction;336
11.9.2;2. The General Plan of the Proof;339
11.9.3;3. The Main Terms;344
11.9.4;4. The Error Terms;347
11.9.5;References;356
11.10;Chapter 18. A New Upper Bound in the Linear Sieve;358
11.10.1;1. Introduction;358
11.10.2;2. A New Upper Bound Method;360
11.10.3;3. Proof of Theorem 1;365
11.10.4;References;368
11.11;Chapter 19. On the Distribution of log|.'(1/2 + it)|;370
11.11.1;1;370
11.11.2;2;371
11.11.3;3;372
11.11.4;4;375
11.11.5;5;376
11.11.6;6;377
11.11.7;7;383
11.11.8;8;387
11.11.9;9;395
11.11.10;10;395
11.11.11;11;395
11.11.12;12;396
11.11.13;References;396
11.12;Chapter 20. Selberg's Lower Bound of the First Eigenvalue for Congruence Groups;398
11.12.1;1. Introduction;398
11.12.2;2. An Estimate for Sums of Kloosterman Sums;399
11.12.3;3. Kloosterman Sums Formula;400
11.12.4;4 Proof of Selberg's Bound;401
11.12.5;References;402
11.13;Chapter 21.
Discrete Subgroups and Ergodic Theory;404
11.13.1;1. Formulation of Results and Conjectures;404
11.13.2;2. Proof of Theorem 2;411
11.13.3;3. Proofs of Lemmas 5–7;418
11.13.4;4. Proof of Theorem 1';422
11.13.5;References;424
11.14;Chapter 22. Good Rational Approximation Derived from Thue's Inequality;426
11.14.1;1. Introduction;426
11.14.2;2. On the Proof of Proposition 2;430
11.14.3;References;435
11.15;Chapter 23. The Selberg Zeta-Function of a Kleinian Group;436
11.15.1;1. Introduction;436
11.15.2;2. Geometrical Considerations;440
11.15.3;3. Eisenstein Series;445
11.15.4;4. The Asymptotic Analysis;454
11.15.5;5. Main Theorem;462
11.15.6;References;467
11.16;Chapter 24.
Nonarithmetic Lattices in Lobachevsky Spaces of Arbitrary Dimension;470
11.17;Chapter 25.
On Some Functions Connected with the Sieve;472
11.17.1;1;472
11.17.2;2;474
11.17.3;3;475
11.17.4;4;477
11.17.5;5;478
11.17.6;6;480
11.17.7;References;482
11.18;Chapter 26. Special Values of Selberg's Zeta- function;484
11.18.1;References;492
11.19;Chapter 27.
Sifting Problems, Sifting Density, and Sieves;494
11.19.1;References;511
11.20;Chapter 28. Remarks on the Sieving Limit of the Buchstab—Rosser Sieve;512
11.20.1;1. Introduction;512
11.20.2;2. Some Preparations;513
11.20.3;3. Some Lemmas;515
11.20.4;4. Proof that gk(s) > 0 for . = t;520
11.20.5;5. Proof that gk(s) changes sign for . ˜ t – O(k-2/3);521
11.20.6;6. Estimation of ß;527
11.20.7;References;529
11.21;Chapter 29.
Recent Work on Waring's Problem;530
11.21.1;1. Introduction;530
11.21.2;2. Waring's Problem;530
11.21.3;3. Diagonal Forms;532
11.21.4;4. An Exponential Sum;532
11.21.5;5. Description of the Method;534
11.21.6;6. A Fundamental Lemma;535
11.21.7;References;536