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.

ContributorsParticipants in the Selberg SymposiumPrefaceForewordHistorical Introduction 1. Prehistory of the Zeta-FunctionSurvey Lectures on Selberg'S Work 2. The Trace Formula and Hecke Operators 3. Selberg's Sieve and Its Applications 4. The Rankin-Selberg Method: A Survey 5. On the Base Change Problem: After J. Arthur and L. Clozel 6. Eisenstein Series, the Trace Formula, and the Modern Theory of Automorphic Forms 7. Selberg's Work on the Zeta Function 8. Selberg's Work on the Arithmeticity of Lattices and Its RamificationsResearch Announcements 9. Mean Values of the Riemann Zeta Function with Application to the Distribution of Zeros 10. Geometric Ramanujan Conjecture and Drinfeld Reciprocity Law 11. On the Brun-Titchmarsh Theorem 12. A Double Sum over Primes and Zeros of the Zeta Function 13. Integral Representations of Eisenstein Series and L-Functions 14. Recent Results on Automorphic L-Functions 15. Explicit Formulae as Trace Formulae 16. Some Remarks on the Sieve Method 17. Critical Zeros of GL(2) L-Functions 18. A New Upper Bound in the Linear Sieve 19. On the Distribution of log |¿'(1/2 + it)| 20. Selberg's Lower Bound of the First Eigenvalue for Congruence Groups 21. Discrete Subgroups and Ergodic Theory 22. Good Rational Approximation Derived from Thue's Inequality 23. The Selberg Zeta Function of a Kleinian Group 24. Nonarithmetic Lattices in Lobachevsky Spaces of Arbitrary Dimension 25. On Some Functions Connected with the Sieve 26. Special Values of Selberg's Zeta Function 27. Sifting Problems, Sifting Density and Sieves 28. Remarks on the Sieving Limit of the Buchstab-Rosser Sieve 29. Recent Work on Waring's Problem