Don’t let the friendly word “invitation” in the title fool you. This is a great book, but not one for the mathematically faint of heart. The book grew from undergraduate research seminars at Princeton, NYU, Ohio State, and Brown and ranges widely over elementary, algebraic, and analytic number theory. The principal aims of the book are to show connections between seemingly different topics and to expose undergraduates to current research topics. Even though expositions of some mathematical prerequisites/corequisites are provided in the text, a reader who hopes to benefit from the entire book would be well advised to have a background in calculus, probability and statistics, linear algebra, and complex analysis.
The eighteen chapters of this book are divided into five parts: Part 1 is devoted to basic number theory (including cryptography, the Riemann zeta function, some complex function theory, Dirichlet characters, and L-functions); Part 2 deals with approximating numbers with rationals (including a proof of Roth’s Theorem on the approximation of real algebraic numbers and a treatment of continued fractions); Part 3 covers probabilistic methods and equidistribution (including hypothesis testing and Fourier analysis); Part 4 describes the Circle Method and some of its applications, especially the distribution of Germain primes (p such that both p and (p – 1)/2 are prime); Part 5 investigates random matrix theory and L-functions. There are two appendices reviewing key ideas in analysis and linear algebra. Two additional appendices contain hints/comments on the exercises and “Concluding Remarks.” There is a twenty-page Bibliography containing books, published papers, preprints, and course notes. At the beginning of the Bibliography, a URL is given which provides links to many of the references. The Index is very helpful and includes an entry for “techniques” (absolute value squared, …, Method of Descent, …, Poisson Summation, …,weighted prime sums). The authors suggest that “a typical semester class would cover material from one part of the book (as well as whatever background material is needed)” — assuming some familiarity with the contents of Part 1.
Typically in this book, a topic is introduced briefly, usually with motivational and/or historical comments and references to deeper treatments. There are frequent exercises integrated into the text, and “Research Projects” are spread throughout. For example, on p. 15, after a brief introduction to congruences, the reader is asked to prove the Chinese Remainder Theorem as an exercise. The first research project, on pages 18–19, asks for investigations of the spacings between adjacent Carmichael numbers (composites n such that for all positive integers a, a to the power n – 1 is congruent to 1 mod n). To me the last part of this book was the least familiar and most illustrative of the philosophy of the entire book. This spirit of exploration and interconnectivity is captured in the following remark: “In some sense, the zeros of L-functions behave like the eigenvalues of matrices which in turn behave like the energy levels of heavy nuclei.”
Although I can’t agree that this book is “accessible to beginning undergraduates,” as the back cover blurb states, I do claim that it is a fine book for talented and mathematically mature undergraduates, for graduate students, and for anyone looking for information on modern number theory.
Henry Ricardo (firstname.lastname@example.org) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.