Contents ix

§7.5. The PNT for arithmetic progressions 198

§7.6. Notes 205

Exercises 205

Chapter 8. Mainly Analysis 209

§8.1. The Poisson summation formula 209

§8.2. Theta functions 216

§8.3. The gamma function 223

§8.4. The functional equation of ζ(s) 227

§8.5. The functional equation of L(s, χ) 231

§8.6. The Hadamard factorization theorem 235

§8.7. The Phragm´ en-Lindel¨ of principle 240

§8.8. Notes 243

Exercises 247

Chapter 9. Euler Products and Number Fields 255

§9.1. The Dedekind zeta function 255

§9.2. The analytic class number formula 262

§9.3. Class numbers of quadratic fields 269

§9.4. A discriminant bound 275

§9.5. The Prime Ideal Theorem 281

§9.6. A proof of the Ikehara theorem 287

§9.7. Induced representations 293

§9.8. Artin L-functions 296

§9.9. Notes 302

Exercises 303

Chapter 10. Explicit Formulas 307

§10.1. The von Mangoldt formula 307

§10.2. The primes and RH 314

§10.3. The Guinand-Weil formula 315

§10.4. Notes 322

Exercises 324

Chapter 11. Supplementary Exercises 327

Exercises 327

Solutions 330