vi Contents
§9.8. The Dickman function 147
§9.9. Consecutive smooth numbers 149
Problems on Chapter 9 150
Chapter 10. The Hardy-Ramanujan and Landau Theorems 157
§10.1. The Hardy-Ramanujan inequality 157
§10.2. Landau’s theorem 159
Problems on Chapter 10 164
Chapter 11. The abc Conjecture and Some of Its Applications 167
§11.1. The abc conjecture 167
§11.2. The relevance of the condition ε 0 168
§11.3. The Generalized Fermat Equation 171
§11.4. Consecutive powerful numbers 172
§11.5. Sums of k-powerful numbers 172
§11.6. The Erd˝ os-Woods conjecture 173
§11.7. A problem of Gandhi 174
§11.8. The k-abc conjecture 175
Problems on Chapter 11 176
Chapter 12. Sieve Methods 179
§12.1. The sieve of Eratosthenes 179
§12.2. The Brun sieve 180
§12.3. Twin primes 184
§12.4. The Brun combinatorial sieve 187
§12.5. A Chebyshev type estimate 187
§12.6. The Brun-Titchmarsh theorem 188
§12.7. Twin primes revisited 190
§12.8. Smooth shifted primes 191
§12.9. The Goldbach conjecture 192
§12.10. The Schnirelman theorem 194
§12.11. The Selberg sieve 198
§12.12. The Brun-Titchmarsh theorem from the Selberg sieve 201
§12.13. The Large sieve 202
§12.14. Quasi-squares 203
§12.15. The smallest quadratic nonresidue modulo p 204
Problems on Chapter 12 206
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