Jeffrey Stopple

This undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. The author pays special attention to the rich history of the subject and ancient questions on polygonal numbers, perfect numbers and amicable pairs, as well as to the important open problems (some of which have million dollar prizes). The capstone of the book is a brief presentation of the Riemann zeta function, which determines the distribution of prime numbers, and of the significance of the Riemann Hypothesis.

Giuliana Davidoff, Peter Sarnak, Alain Valette

This text is a self contained treatment of expander graphs and in particular their explicit construction. Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering. The reader needs only a background in elementary algebra, analysis and combinatorics; the authors supply the necessary background from graph theory, number theory, group theory and representation theory. Thus the text can be used as a brief introduction to these subjects and their synthesis in modern mathematics.

Edited by Henri Darmon, Shou-wu Zhang

The seminal formula of Gross and Zagier has led to many generalisations and extensions in a variety of different directions, spawning a fertile area of study that remains active today. This volume, based on a workshop on Special Values of Rankin L-series held at the MSRI in December 2001, presents thirteen articles written by many of the leading contributors in the field, on the history of the Gross-Zagier formula and recent developments. Topics include the theory of complex multiplication, automorphic forms, the Rankin-Selberg method, arithmetic intersection theory, and Iwasawa theory.

George E. Andrews, Kimmo Eriksson

The theory of integer partitions is a subject of enduring interest. A major research area in its own right, it has found numerous applications, and celebrated results such as the Rogers-Ramanujan identities make it a topic filled with the true romance of mathematics. The aim in this introductory textbook is to provide an accessible and wide ranging introduction to partitions, without requiring anything more of the reader than some familiarity with polynomials and infinite series. Many exercises are included, together with some solutions and helpful hints.

J. W. S. Cassels

Elliptic curves constitute one of the oldest and liveliest centres of research in number theory today. This book, which is addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. Many examples and exercises are included for the reader. For those new to elliptic curves, whether they are graduate students or specialists from other fields, this will be a fine introductory text.

Daniel Bump

The theory of automorphic forms is a cornerstone in modern number theory. It was an essential ingredient of the proof of Fermat's Last Theorem. However, in large part because of the lack of a suitable text, this theory has been difficult for students to learn. This book addresses that difficulty. It begins with extensive foundational material and builds to topics on the research frontier, such as the Langlands conjectures and the Weil representation. Researchers as well as students will find this a valuable guide to a notoriously difficult subject.

Sergei Konyagin, Igor Shparlinski

The theme of this book is the study of the distribution of integer powers modulo a prime number. It provides numerous new links between number theory and computer science as well as other areas of mathematics. Possible applications include (but are not limited to) complexity theory, random number generation, cryptography, and coding theory. The main method discussed is based on bounds of exponential sums and the book contains many estimates of these, including new estimates of classical Gaussian sums. It also contains many open questions and proposals for further research.

I. Blake, G. Seroussi, N. Smart

This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.

V. I. Bernik, M. M. Dodson

This book is concerned with Diophantine approximation on smooth manifolds embedded in Euclidean space, and its aim is to develop a coherent body of theory comparable with that which already exists for classical Diophantine approximation. In particular, this book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. All researchers with an interest in Diophantine approximation will welcome this book.