Contents ix

§6. An application to the Goldbach problem 196

Notes 201

Exercises 202

Chapter 7. An Elementary Proof of the Prime Number Theorem 213

§1. Introduction 214

§2. Chebyshev’s theorems revisited 217

§3. Proof of Selberg’s fundamental formula 221

§4. Removing the explicit appearance of primes 224

§5. Nevanlinna’s finishing strategy 231

Notes 235

Exercises 237

Chapter 8. Perfect Numbers and their Friends 247

§1. Introduction and overview 248

§2. Proof of Dickson’s finiteness theorem 253

§3. How rare are odd perfect numbers? 255

§4. The distribution function of σ(n)/n 259

§5. Sociable numbers 263

Notes 267

Exercises 269

References 279

Index 301