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
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