viii Contents

§6. A cyclotomic proof of quadratic reciprocity 61

§7. Jacobi’s cubic reciprocity law 64

Notes 75

Exercises 77

Chapter 3. Elementary Prime Number Theory, II 85

§1. Introduction 85

§2. The set of prime numbers has density zero 88

§3. Three theorems of Chebyshev 89

§4. The work of Mertens 95

§5. Primes and probability 100

Notes 104

Exercises 107

Chapter 4. Primes in Arithmetic Progressions 119

§1. Introduction 119

§2. Progressions modulo 4 120

§3. The characters of a finite abelian group 123

§4. The L-series at s = 1 127

§5. Nonvanishing of L(1,χ) for complex χ 128

§6. Nonvanishing of L(1,χ) for real χ 132

§7. Finishing up 133

§8. Sums of three squares 134

Notes 139

Exercises 141

Chapter 5. Interlude: A Proof of the Hilbert–Waring Theorem 151

§1. Introduction 151

§2. Proof of the Hilbert–Waring theorem (Theorem 5.1) 152

§3. Producing the Hilbert–Dress identities 156

Notes 161

Chapter 6. Sieve Methods 163

§1. Introduction 163

§2. The general sieve problem: Notation and preliminaries 169

§3. The sieve of Eratosthenes–Legendre and its applications 170

§4. Brun’s pure sieve 175

§5. The Brun–Hooley sieve 182