An Introduction to the Circle Method

Contents
Preface
Index of notations
Chapter 1. Introduction and overview
1.1. Introduction
1.2. Preparatory chapters
1.3. Early developments in the study of Waring’s problem
1.4. The method of exponential sums
1.5. Origins of the circle method and applications to additive problems
Chapter 2. Fundamental theorem of arithmetic
2.1. Mathematical induction
2.2. Divisibility
2.3. Greatest common divisor
2.4. Prime numbers and unique factorization
Chapter 3. Arithmetic functions
3.1. Multiplicative functions
3.2. Möbius function and Möbius inversion
3.3. Greatest integer function
3.4. The big-Ox and little-ox notations
3.5. Averages of arithmetical functions
3.6. Technique of partial summation
3.7. The Cauchy–Schwarz and Hölder inequalities
Chapter 4. Introduction to congruence arithmetic
4.1. Definition and basic properties of congruences
4.2. Congruence powers and Euler’s theorem
4.3. Linear congruence equations
4.4. Linear congruences and the Chinese remainder theorem
4.5. Polynomial congruences
4.6. Order and primitive roots
Chapter 5. Distribution of prime numbers
5.1. Dirichlet series
5.2. Euler products and Dirichlet series
5.3. Analytic properties of Dirichlet series
5.4. Distribution functions for prime numbers
5.5. Primes in arithmetic progressions
5.6. Dirichlet characters and Dirichlet ?-functions
5.7. Ramanujan sums and Ramanujan series
Chapter 6. An introduction to Waring’s problem
6.1. Fermat’s two square theorem
6.2. Lagrange’s four square theorem
6.3. A conjectured value for ?(?)
6.4. The easier Waring’s problem
Chapter 7. Waring’s problem
7.1. Schnirelmann density
7.2. Schnirelmann density and Waring’s problem
7.3. Proof of Linnik’s theorem
Chapter 8. Exponential sums
8.1. Exponential sums for polynomials of degree 1
8.2. Exponential sums and Diophantine approximation
8.3. Exponential sums over primes
Chapter 9. The circle method and Waring’s problem
9.1. An outline of the circle method
9.2. The contribution from the major arcs
9.3. The singular integral
9.4. Singular series
9.5. Minor arcs in Waring’s problem
Chapter 10. The circle method and the Goldbach conjectures
10.1. Major and minor arcs
10.2. Contribution from the major arcs
10.3. Contribution from the minor arcs
10.4. Comments about Vinogradov’s theorem
10.5. The circle method and the binary Goldbach conjecture
Chapter 11. Epilogue
11.1. The philosophy of the circle method
11.2. An axiomatic framework
11.3. The singular series
11.4. The minor arcs
11.5. The future of the circle method
Bibliography
Index
Back cover