viii Contents
Chapter 3. Characters and Euler Products 59
§3.1. The Euler product formula 59
§3.2. Convergence of Dirichlet series 64
§3.3. Harmonics 67
§3.4. Group representations 71
§3.5. Fourier analysis on finite groups 76
§3.6. Primes in arithmetic progressions 83
§3.7. Gauss sums and primitive characters 89
§3.8. The character group 95
§3.9. Notes 99
Exercises 103
Chapter 4. The Circle Method 111
§4.1. Diophantine equations 111
§4.2. The major arcs 116
§4.3. The singular series 123
§4.4. Weyl sums 130
§4.5. An asymptotic estimate 138
§4.6. Notes 144
Exercises 150
Chapter 5. The Method of Contour Integrals 157
§5.1. The Perron formula 157
§5.2. Bounds for Dirichlet L-functions 162
§5.3. Notes 165
Exercises 166
Chapter 6. The Prime Number Theorem 169
§6.1. A zero-free region 169
§6.2. A proof of the PNT 173
§6.3. Notes 177
Exercises 179
Chapter 7. The Siegel-Walfisz Theorem 183
§7.1. Zero-free regions for L-functions 183
§7.2. An idea of Landau 190
§7.3. The theorem of Siegel 193
§7.4. The Borel-Carath´ eodory lemma 196
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