Equidistribution in Number Theory, An Introduction

- Preface

- Contributors

- Biographical Sketches of the Lecturers

- Uniform Distribution

1. Uniform Distribution mod One.
2. Fractional Parts of an2 .
3. Uniform Distribution mod N.
4. Normal Numbers. Sieving and the Erdos—Kac Theorem

Uniform Distribution, Exponential Sums, and Cryptography
1. Randomness and Pseudorandomness
2. Uniform Distribution and Exponential Sums.
3. Exponential Sums and Cryptography.
4. Some Exponential Sum Bounds.
5. General Modulus and Discrepancy of Diffie—Hellman Triples.
6. Pseudorandom Number Generation
7. Large Periods and the Carmichael Function
8. Exponential Sums to General Modulus
9. Sums over Elliptic Curves
10. Proof Sketch of Theorem 4

1. The Distribution of Prime Numbers
1. The Cramer Model and Gaps Between Consecutive Primes
2. The Distribution of Primes in Longer Intervals
3. Maier's Method and an "Uncertainty Principle"

Torsion Points on Curves

1. Introduction
2. A Proof Using Galois Theory
3. Polynomials Vanishing at Roots of Unity

The distribution of roots of a polynomial
1. Introduction. 2 Algebraic Numbers
3. In k Dimensions: the Bilu Equidistribution Theorem.
4. Lower Bounds on Heights.
5. Compact Sets with Minimal Energy

Manin—Mumford, Andr —Oort, the Equidistribution Point of View
1. Introduction
2. Informal Examples of Equi-Distribution.
3. The Manin—Mumford and the Andr —Oort Conjecture.
4. Equidistribution of Special Subvarieties

Analytic Methods for the Distribution of Rational Points on Algebraic Varieties

1. Introduction to the Hardy—Littlewood Circle Method.
2. Major Arcs and Local Factors in the Hardy—Littlewood Circle Method.
3. The Minor Arcs in the Hardy—Littlewood Circle Method.
4. Combining Analytic and Geometric Methods

Universal Torsors over Del Pezzo Surfaces and Rational Points

1. Introduction.
2. Geometric Background.
3. Manin's Conjecture.
4. The Universal