Contents v
§6.3. Multiplicative functions 79
§6.4. Generating functions and Dirichlet products 81
§6.5. Wintner’s theorem 82
§6.6. Additive functions 85
§6.7. The average orders of ω(n) and Ω(n) 86
§6.8. The average order of an additive function 87
§6.9. The Erd˝ os-Wintner theorem 88
Problems on Chapter 6 89
Chapter 7. The Local Behavior of Arithmetic Functions 93
§7.1. The normal order of an arithmetic function 93
§7.2. The Tur´ an-Kubilius inequality 94
§7.3. Maximal order of the divisor function 99
§7.4. An upper bound for d(n) 101
§7.5. Asymptotic densities 103
§7.6. Perfect numbers 106
§7.7. Sierpi´ nski, Riesel, and Romanov 106
§7.8. Some open problems of an elementary nature 108
Problems on Chapter 7 109
Chapter 8. The Fascinating Euler Function 115
§8.1. The Euler function 115
§8.2. Elementary properties of the Euler function 117
§8.3. The average order of the Euler function 118
§8.4. When is φ(n)σ(n) a square? 119
§8.5. The distribution of the values of φ(n)/n 121
§8.6. The local behavior of the Euler function 122
Problems on Chapter 8 124
Chapter 9. Smooth Numbers 127
§9.1. Notation 127
§9.2. The smallest prime factor of an integer 127
§9.3. The largest prime factor of an integer 131
§9.4. The Rankin method 137
§9.5. An application to pseudoprimes 141
§9.6. The geometric method 145
§9.7. The best known estimates on Ψ(x, y) 146
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