1.3 Big O, little o, and $\sim$ notation 12
1.4 The radius of convergence and $RT_\rho$ 13
1.5 Growth rate of coefficients 15
Chapter 2. Counting Functions, Fundamental Identities 17
2.1 Defining additive number systems 17
2.2 Examples of additive number systems 21
2.3 Counting functions, fundamental identities 25
2.4 Global counts 33
2.5 Alternate version of the fundamental identity 34
2.6 Reduced additive number systems 37
2.7 Finitely generated number systems 39
2.8 $a^\star (n)$ is eventually positive 43
Chapter 3. Density and Partition Sets 45
3.1 Asymptotic density 45
3.2 Dirichlet density 48
3.3 The standard assumption 51
3.4 The set of additives of an element 51
3.5 Partition sets 54
3.6 Generating series of partition sets 57
3.7 Partition sets have Dirichlet density 58
3.8 Schur's Tauberian Theorem 62
3.9 Simple partition sets 67
3.10 The asymptotic density of $\gamma P$ 68
3.11 Adding an indecomposable 70
Chapter 4. The Case $\rho = 1$ 75
4.1 The fundamental results when $\rho = 1$ 75
4.2 The Bateman and Erd\"os Result 77
4.3 Stewart's Sum Theorem 83
4.4 $\rho = 1$ does not imply $RT_1$ 84
4.5 Further Results on $RT_1$ 85
Chapter 5. The Case 0ドル< \rho < 1$ 87
5.1 Compton's Tauberian Theorem 87
5.2 When is $\delta (B) = 0$? 90
5.3 The density of $\overline{B}$ 91
5.4 Compton's Density Theorem 92
5.5 The Knopfmacher, Knopfmacher, Warlimont asymptotics 93
Chapter 6. Monadic Second-Order Limit Laws 103
6.1 First-order logic 103
6.2 Monadic second-order logic 104
6.3 Asymptotic density of subsets of $K$ 106
6.4 Quantifier rank and equivalent formulas 107
6.5 Ehrenfeucht-Fraiss\'e games 109
6.6 Adequate classes of structures 119
Part 2. Multiplicative Number Systems 125
Chapter 7. Background from Analysis 127
7.1 Series and infinite products 127
7.2 Dirichlet series expansions 133
7.3 The abcissa of convergence and $RV_\alpha $ 136
7.4 Growth rate of coefficients 140
Chapter 8. Counting Functions and Fundamental Identities 143
8.1 Defining multiplicative number systems 143
8.2 Examples of multiplicative number systems 144
8.3 Counting functions, fundamental identities 145
8.4 Alternate version of the fundamental identity 153
8.5 Finitely generated multiplicative number systems 156
Chapter 9. Density and Partition Sets 159
9.1 Asymptotic density 159
9.2 Dirichlet density 160
9.3 The standard assumption 164
9.4 The set of multiples of an element 164
9.5 Partition sets 166
9.6 Generating series of partition sets 169
9.7 Partition sets have Dirichlet density 170
9.8 Discrete multiplicative number systems 174
9.9 When sets $bA$ have global asymptotic density 177
9.10 The strictly multiplicative case and $RV_\alpha$ 181
9.11 The discrete case and $RV_\alpha$ 181
9.12 Analog of Schur's Tauberian Theorem 182
9.13 Simple partition sets 188
9.14 The asymptotic density of $P^\gamma$ 189
9.15 Adding an indecomposable 191
9.16 First conjecture 194
Chapter 10. The Case $\alpha = 0$ 195
10.1 The fundamental results when $\alpha = 0$ 195
10.2 Odlyzko's Product Theorem 196
10.3 Further results on $RV_0$ 198
10.4 Second conjecture 199
Chapter 11. The Case 0ドル < \alpha < \infty $ 201
11.1 Analog of Compton's Tauberian Theorem 201
11.2 When is $\Delta (B) = 0$? 204
11.3 The density of $\overline{B}$ 205
11.4 S\'ark\"ozy's Density Theorem 206
11.5 Generalizing Oppenheim's asymptotics 208
11.6 Third conjecture 216
Chapter 12. First-Order Limit Laws 217
12.1 Asymptotic density of subsets of $K$ 217
12.2 The Feferman-Vaught Theorem 218
12.3 Skolem's analysis of the first-order calculus of classes 223
12.4 $K_\phi$ is a disjoint union of partition classes 226
12.5 Applications 227
12.6 Finite dimensional structures 231
12.7 The main problem 231
Appendix A. Formal Power Series 233
Appendix B. Refined Counting 251
Appendix C. Consequences of $\delta (P) = 0$ 261
Appendix D. On the Monotonicity of $a(n)$ When
$p(n) \leq 1$ 269
Appendix E. Results of Woods 273