Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing. Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers The user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.

This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry. Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research. This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.

This set of lectures provides a structured introduction to the concept of equidistribution in number theory. This concept is of growing importance in many areas, including cryptography, zeros of L-functions, Heegner points, prime number theory, the theory of quadratic forms, and the arithmetic aspects of quantum chaos.

The volume brings together leading researchers from a range of fields who reveal fascinating links between seemingly disparate areas.

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz's Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites. Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called "the plaid model," has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.

Die Kryptologie, eine jahrtausendealte "Geheimwissenschaft," gewinnt zusehends praktische Bedeutung fur den Schutz von Kommunikationswegen, Datenbanken und Software. Neben ihre Nutzung in rechnergestutzten offentlichen Nachrichtensystemen ("public keys") treten mehr und mehr rechnerinterne Anwendungen, wie Zugriffsberechtigungen und der Quellenschutz von Software. - Der erste Teil des Buches behandelt die Geheimschriften und ihren Gebrauch - die Kryptographie. Dabei wird auch auf das aktuelle Thema "Kryptographie und Grundrechte des Burgers" eingegangen. Im zweiten Teil wird das Vorgehen zum unbefugten Entziffern einer Geheimschrift - die Kryptanalyse - besprochen, wobei insbesondere Hinweise zur Beurteilung der Verfahrenssicherheit gegeben werden. Mit der vorliegenden dritten Auflage wurde das Werk auf den neuesten Stand gebracht. - Das Buch setzt nur mathematische Grundkenntnisse voraus. Mit einer Fulle spannender, lustiger und bisweilen anzuglicher Geschichten aus der historischen Kryptologie gewurzt, ist es auch fur Laien reizvoll zu lesen."

The decomposition of the space L2 (G(Q)\G(/A)), where G is a reductive group defined over (Q and /A is the ring of adeles of (Q, is a deep problem at the intersection of number and group theory. Langlands reduced this decomposition to that of the (smaller) spaces of cuspidal automorphic forms for certain subgroups of G. The present book describes this proof in detail. The starting point is the theory of automorphic forms, which can also serve as a first step towards understanding the Arthur-Selberg trace formula. To make the book reasonably self-contained, the authors have also provided essential background to subjects such as automorphic forms, Eisenstein series, Eisenstein pseudo-series (or wave-packets) and their properties. It is thus also an introduction, suitable for graduate students, to the theory of automorphic forms, written using contemporary terminology. It will be welcomed by number theorists, representation theorists, and all whose work involves the Langlands program.

A deep understanding of prime numbers is one of the great challenges in mathematics. In this new edition, fundamental theorems, challenging open problems, and the most recent computational records are presented in a language without secrets. The impressive wealth of material and references will make this book a favorite companion and a source of inspiration to all readers.

Paulo Ribenboim is Professor Emeritus at Queen's University in Canada, Fellow of the Royal Society of Canada, and recipient of the George Polya Award of the Mathematical Association of America. He is the author of 13 books and more than 150 research articles.

From the reviews of the First Edition:

Number Theory and mathematics as a whole will benefit from having such an accessible book exposing advanced material. There is no question that this book will succeed in exciting many new people to the beauty and fascination of prime numbers, and will probably bring more young people to research in these areas. (Andrew Granville, Zentralblatt)"

The legacy of Helmut Hasse, consisting of letters, manuscripts and other - pers, is kept at theHandschriftenabteilung of the University Library at Gottin- ] gen.Hassehadanextensivecorrespondence;helikedtoexchangemathematical ideas, results and methods freely with his colleagues. There are more than 8000 documents preserved. Although not all of them are of equal mathematical - terest, searching through this treasure can help us to assess the development of Number Theory through the 1920's and 1930's. Unfortunately, most of the correspondence is preserved on one side only, i.e., the letterssenttoHasse are availablewhereasmanyoftheletterswhichhadbeensentfromhim, oftenha- written, seem to be lost. So we have to interpolate, as far as possible, from the repliestoHasseandfromothercontexts, inorderto?ndoutwhathehadwritten 1 in his outgoing letters. The present article is largely based on the letters and other documents which I have found concerning the Brauer-Hasse-NoetherTheorem in the theory of algebras; this covers the years around 1931. Besides the do- ments from the Hasse and the Brauer legacy in Gottingen, ] I shall also use some letters from Emmy Noether to Richard Brauer which are preserved at the Bryn Mawr College Library (Pennsylvania, USA)."