Number Theory for the Millennium I
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Building on the tradition of an outstanding series of conferences at the University of Illinois at Urbana-Champaign, the organizers attracted an international group of scholars to open the new Millennium with a conference that reviewed the current state of number theory research and pointed to future directions in the field. The conference was the largest general number theory conference in recent history, featuring a total of 159 talks, with the plenary lectures given by George Andrews, Jean Bourgain, Kevin Ford, Ron Graham, Andrew Granville, Roger Heath-Brown, Christopher Hooley, Winnie Li, Kumar Murty, Mel Nathanson, Ken Ono, Carl Pomerance, Bjorn Poonen, Wolfgang Schmidt, Chris Skinner, K. Soundararajan, Robert Tijdeman, Robert Vaughan, and Hugh Williams. The Proceedings Volumes of the conference review some of the major number theory achievements of this century and to chart some of the directions in which the subject will be heading during the new century. These volumes will serve as a useful reference to researchers in the area and an introduction to topics of current interest in number theory for a general audience in mathematics.
Table of Contents
1. Small Fractional Parts of Quadratic Forms 2. Completeness Problems and the Riemann Hypothesis: An Annotated Bibliography 3. Arbitrarily Tight Bounds on the Distribution of Smooth Integers 4. Cyclotomic Points on Curves 5. On the Distribution of Dirichlet Sums, II 6. Theta Functions and Singular Torsion on Elliptic Curves 7. Mahler’s Measure and Invariants of Hyperbolic Manifolds 8. On the Diophantine Equation (x — 1)/(x — 1) = yq with Negative : X 9. A Dirichlet Series Related to Eigenvalues of the Laplacian for Congruence Subgroups 10. Convolution Operators and Entire Functions with Simple Zeros 11. Ramanujan’s Elliptic Functions to Alternative Bases and Approximations to π 12. Binary Quadratic Forms and the Eta Function 13. Discriminants and Arakelov Euler Characteristics 14. On Sparse Sets Hitting Linear Forms 15. A Survey on Pure and Mixed Exponential Sums Modulo Prime Powers 16. On the Frequency of Vanishing of Quadratic Twists of Modular L-Functions 17. Partition Polynomials: Zero Distribution Problems 18. A Comparison of Two Sieves, II 19. Bernoulli Numbers and Conﬂuent Hypergeometric Functions 20. Three—dimensional Galois Representations with Conjectural Connections to Arithmetic Cohomology 21. A New Approach to Congruences of Kummer Type for Bernoulli Numbers 22. The Least Prime Primitive Root and Linnik’s Theorem 23. Some Remarks on the Additive Structure of the Set of Primes 24. Twisted Hyper-Kloosterman Sums Over Finite Rings of Integers 25. Wirsing Systems and Resultant Inequalities
"The conference was one of the most important international meetings devoted to number theory . . . . Consequently the interested reader finds here not only surveys on the most important contributions to number theory and its applications, but also surveys on methods and techniques used in this important branch of mathematics." -EMS Newsletter, December 2004