Joel Spencer 
Asymptopia [PDF ebook] 

This beautiful book is about how to estimate large quantities-and why. Building on nothing more than first-year calculus, it goes all the way into deep asymptotical methods and shows how these can be used to solve problems in number theory, combinatorics, probability, and geometry. The author is a master of exposition: starting from such a simple fact as the infinity of primes, he leads the reader through small steps, each carefully motivated, to many theorems that were cutting-edge when discovered, and teaches the general methods to be learned from these results. -Laszlo Lovasz, Eoetvoes-Lorand University This is a lovely little travel guide to a country you might not even have heard about – full of wonders, mysteries, small and large discoveries … and in Joel Spencer you have the perfect travel guide! -Guenter M. Ziegler, Freie Universitaet Berlin, coauthor of "Proofs from THE BOOK" Asymptotics in one form or another are part of the landscape for every mathematician. The objective of this book is to present the ideas of how to approach asymptotic problems that arise in discrete mathematics, analysis of algorithms, and number theory. A broad range of topics is covered, including distribution of prime integers, Erdos Magic, random graphs, Ramsey numbers, and asymptotic geometry. The author is a disciple of Paul Erdos, who taught him about Asymptopia. Primes less than $n$, graphs with $v$ vertices, random walks of $t$ steps-Erdos was fascinated by the limiting behavior as the variables approached, but never reached, infinity. Asymptotics is very much an art. The various functions $n/ln n$, $n^2$, $/frac{/ln n}{n}$, $/sqrt{/ln n}$, $/frac{1}{n/ln n}$ all have distinct personalities. Erdos knew these functions as personal friends. It is the author’s hope that these insights may be passed on, that the reader may similarly feel which function has the right temperament for a given task. This book is aimed at strong undergraduates, though it is also suitable for particularly good high school students or for graduates wanting to learn some basic techniques. Asymptopia is a beautiful world. Enjoy!