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