Mathematical Surveys and Monographs 2001; 289 pp; hardcover Volume: 86 ISBN10: 0821826662 ISBN13: 9780821826669 List Price: US$80 Member Price: US$64 Order Code: SURV/86
 This book shows how a study of generating series (power series in the additive case and Dirichlet series in the multiplicative case), combined with structure theorems for the finite models of a sentence, lead to general and powerful results on limit laws, including \(0  1\) laws. The book is unique in its approach to giving a combined treatment of topics from additive as well as from multiplicative number theory, in the setting of abstract number systems, emphasizing the remarkable parallels in the two subjects. Much evidence is collected to support the thesis that local results in additive systems lift to global results in multiplicative systems. All necessary material is given to understand thoroughly the method of Compton for proving logical limit laws, including a full treatment of EhrenfeuchtFraissé games, the FefermanVaught Theorem, and Skolem's quantifier elimination for finite Boolean algebras. An intriguing aspect of the book is to see so many interesting tools from elementary mathematics pull together to answer the question: What is the probability that a randomly chosen structure has a given property? Prerequisites are undergraduate analysis and some exposure to abstract systems. Readership Graduate students and research mathematicians interested in combinatorics, number theory and logic. Reviews "Shows an exciting connection between combinatorics, number theory and logic, and certainly deserves to be more widely known. The book gives a very clear account of it and it is easily readable."  Zentralblatt MATH "This book is a lucid, selfcontained introduction to a fascinating interaction between analysis, combinatorics, logic and number theory ... accessible to an undergraduate and gives interesting examples to illustrate the concepts."  Mathematical Reviews Table of Contents Additive number systems  Background from analysis
 Counting functions, fundamental identities
 Density and partition sets
 The case \(\rho = 1\)
 The case \(0 < \rho < 1\)
 Monadic secondorder limit laws
Multiplicative number systems  Background from analysis
 Counting functions and fundamental identities
 Density and partition sets
 The case \(\alpha = 0\)
 The case \(0 < \alpha < \infty\)
 Firstorder limit laws
 Appendix A. Formal power series
 Appendix B. Refined counting
 Appendix C. Consequences of \(\delta(\mathsf P) = 0\)
 Appendix D. On the monotonicity of \(a(n)\) when \(p(n) \leq 1\)
 Appendix E. Results of Woods
 Bibliography
 Symbol index
 Subject index
