vi CONTENTS
1. Riemann's Explicit Formula 143
2. Von Mangoldt's Explicit Formula 146
3. Delsarte's Explicit Formula 149
4. Weil's Explicit Formulas 152
Notes 175
Chapter V. BOUNDS ON DISCRIMINANTS AND CONDUCTORS 177
1. The Fundamental Principle 183
2. Stark's Inequality 184
3. Odlyzko's Method 189
4. The Method of Explicit Formulas 194
Notes 201
Chapter VI. NON-VANISHING THEOREMS 203
1. The Method of Hadamard and de La Vallee Poussin 204
2. The Ramanujan Conjecture 226
3. The Method of Eisenstein Series 230
4. The Langlands-Shahidi Method 258
5. The Hilbert Space Method 262
Notes 267
Appendix A. THE LOCAL THEORY OF ROOT NUMBERS: A Survey 269
1. Artin L-Functions and their Functional Equations 269
2. The Theory of Local Root Numbers 271
3. The Relations in the Kernel of Brauer Induction 275
Bibliography 283
Index 291
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