N atur non facit saltus? This book is devoted to the fundamental problem which arises contin uously in the process of the human investigation of reality: the role of a mathematical apparatus in a description of reality. We pay our main attention to the role of number systems which are used, or may be used, in this process. We shall show that the picture of reality based on the standard (since the works of Galileo and Newton) methods of real analysis is not the unique possible way of presenting reality in a human brain. There exist other pictures of reality where other num ber fields are used as basic elements of a mathematical description. In this book we try to build a p-adic picture of reality based on the fields of p-adic numbers Qp and corresponding analysis (a particular case of so called non-Archimedean analysis). However, this book must not be considered as only a book on p-adic analysis and its applications. We study a much more extended range of problems. Our philosophical and physical ideas can be realized in other mathematical frameworks which are not obliged to be based on p-adic analysis. We shall show that many problems of the description of reality with the aid of real numbers are induced by unlimited applications of the so called Archimedean axiom."

This book is an authoritative description of the various approaches to and methods in the theory of irregularities of distribution. The subject is primarily concerned with number theory, but also borders on combinatorics and probability theory. The work is in three parts. The first is concerned with the classical problem, complemented where appropriate with more recent results. In the second part, the authors study generalizations of the classical problem, pioneered by Schmidt. Here, they include chapters on the integral equation method of Schmidt and the more recent Fourier transform technique. The final part is devoted to Roth's '1/4-theorem'.

A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke's theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr's theory of equivalence of general Dirichlet series.

This is a systematic account of the multiplicative structure of integers, from the probabilistic point of view. The authors are especially concerned with the distribution of the divisors, which is as fundamental and important as the additive structure of the integers, and yet until now has hardly been discussed outside of the research literature. Hardy and Ramanujan initiated this area of research and it was developed by Erdos in the thirties. His work led to some deep and basic conjectures of wide application which have now essentially been settled. This book contains detailed proofs, some of which have never appeared in print before, of those conjectures that are concerned with the propinquity of divisors. Consequently it will be essential reading for all researchers in analytic number theory.

This is a integrated presentation of the theory of exponential diophantine equations. The authors present, in a clear and unified fashion, applications to exponential diophantine equations and linear recurrence sequences of the Gelfond-Baker theory of linear forms in logarithms of algebraic numbers. Topics covered include the Thue equations, the generalised hyperelliptic equation, and the Fermat and Catalan equations. The necessary preliminaries are given in the first three chapters. Each chapter ends with a section giving details of related results.

Modular forms are functions with an enormous amount of symmetry that play a central role in number theory, connecting it with analysis and geometry. They have played a prominent role in mathematics since the 19th century and their study continues to flourish today. Modular forms formed the inspiration for Langlands' conjectures and play an important role in the description of the cohomology of varieties defined over number fields. This collection of up-to-date articles originated from the conference 'Modular Forms' held on the Island of Schiermonnikoog in the Netherlands. A broad range of topics is covered including Hilbert and Siegel modular forms, Weil representations, Tannakian categories and Torelli's theorem. This book is a good source for all researchers and graduate students working on modular forms or related areas of number theory and algebraic geometry.

This book contains thirty-three papers from among the thirty-eight papers presented at the Fourth International Conference on Fibonacci Numbers and Their Applications which was held at Wake Forest University, Winston-Salem, North Carolina from July 30 to August 3, 1990. These papers have been selected after a careful review by well known referees in the field, and they range from elementary number theory to probability and statistics. The Fibonacci numbers and recurrence relations are their unifying bond. It is anticipated that this book, like its three predecessors, will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications. March 1, 1991 The Editors Gerald E. Bergum South Dakota State University Brookings, South Dakota, U. S. A. Alwyn F. Horadam University of New England Armidale, N. S. W. , Australia Andreas N. Philippou Minister of Education Ministry of Education Nicosia, Cyprus xv THE ORGANIZING COMMITTEES LOCAL COMMITTEE INTERNATIONAL COMMITTEE Howard, Fred T. , Co-Chair Horadam, A. F. (Australia), Co-Chair Waddill, Marcellus E. , Co-Chair Philippou, A. N. (Cyprus), Co-Chair Hayashi, Elmer K. Ando, S. (Japan) Bergum, G. E. (U. S. A. ) Vaughan, Theresa Harrell, Deborah Bicknell-Johnson, M. B. (U. S. A. ) Campbell, Colin (Scotland) Filipponi, Piero (Italy) Kiss, P. (Hungary) Turner, J. C. (New Zealand) xvii LIST OF CONTRIBUTORS TO THE CONFERENCE *ALFORD, CECIL 0. , (coauthor Daniel C. Fielder) "Pascal's Triangle: Top Gun or Just One of the Gang?" *ANDERSON, PETER G. , "A Fibonacci-Based Pseudo-Random Number Generator.

To mark the World Mathematical Year 2000 an International Conference on Number Theory and Discrete Mathematics in honour of the legendary Indian Mathematician Srinivasa Ramanuj~ was held at the centre for Advanced study in Mathematics, Panjab University, Chandigarh, India during October 2-6, 2000. This volume contains the proceedings of that conference. In all there were 82 participants including 14 overseas participants from Austria, France, Hungary, Italy, Japan, Korea, Singapore and the USA. The conference was inaugurated by Prof. K. N. Pathak, Hon. Vice-Chancellor, Panjab University, Chandigarh on October 2, 2000. Prof. Bruce C. Berndt of the University of Illinois, Urbana Chaimpaign, USA delivered the key note address entitled "The Life, Notebooks and Mathematical Contributions of Srinivasa Ramanujan". He described Ramanujan--as one of this century's most influential Mathematicians. Quoting Mark K. ac, Prof. George E. Andrews of the Pennsylvania State University, USA, in his message for the conference, described Ramanujan as a "magical genius". During the 5-day deliberations invited speakers gave talks on various topics in number theory and discrete mathematics. We mention here a few of them just as a sampling: * M. Waldschmidt, in his article, provides a very nice introduction to the topic of multiple poly logarithms and their special values. * C.

the attention of The publication of Charles Pisot's thesis in 1938 brought to the mathematical community those marvelous numbers now known as the Pisot numbers (or the Pisot-Vijayaraghavan numbers). Although these numbers had been discovered earlier by A. Thue and then by G. H. Hardy, it was Pisot's result in that paper of 1938 that provided the link to harmonic analysis, as discovered by Raphael Salem and described in a series of papers in the 1940s. In one of these papers, Salem introduced the related class of numbers, now universally known as the Salem numbers. These two sets of algebraic numbers are distinguished by some striking arith metic properties that account for their appearance in many diverse areas of mathematics: harmonic analysis, ergodic theory, dynamical systems and alge braic groups. Until now, the best known and most accessible introduction to these num bers has been the beautiful little monograph of Salem, Algebraic Numbers and Fourier Analysis, first published in 1963. Since the publication of Salem's book, however, there has been much progress in the study of these numbers. Pisot had long expressed the desire to publish an up-to-date account of this work, but his death in 1984 left this task unfulfilled."

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2002.

The subject of this book is the study of automorphic distributions, by which is meant distributions on R2 invariant under the linear action of SL(2, Z), and of the operators associated with such distributions under the Weyl rule of symbolic calculus.

Researchers and postgraduates interested in pseudodifferential analyis, the theory of non-holomorphic modular forms, and symbolic calculi will benefit from the clear exposition and new results and insights.

This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat's Last Theorem, and Iwasawa's theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott's proof of the vanishing of Iwasawa's f-invariant.

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium senes. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book of Records, reminded me very gently that the most "innumerate" people of the world are of a certain trible in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes, Morris, I'm from Brazil, but my book will contain numbers different from *one.''' He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name) and consists of about 16 million decimal digits of the number Te. "I assure you, Morris, that in spite of the beauty of the appar ent randomness of the decimal digits of Te, I'll be sure that my text will include also some words." And then I proceeded putting together the magic combina tion of words and numbers, which became The Book of Prime Number Records. If you have seen it, only extreme curiosity could impel you to have this one in your hands. The New Book of Prime Number Records differs little from its predecessor in the general planning. But it contains new sections and updated records.

Finslerian Laplacians have arisen from the demands of modelling the modern world. However, the roots of the Laplacian concept can be traced back to the sixteenth century. Its phylogeny and history are presented in the Prologue of this volume. The text proper begins with a brief introduction to stochastically derived Finslerian Laplacians, facilitated by applications in ecology, epidemiology and evolutionary biology. The mathematical ideas are then fully presented in section II, with generalizations to Lagrange geometry following in section III. With section IV, the focus abruptly shifts to the local mean-value approach to Finslerian Laplacians and a Hodge-de Rham theory is developed for the representation on real cohomology classes by harmonic forms on the base manifold. Similar results are proved in sections II and IV, each from different perspectives. Modern topics treated include nonlinear Laplacians, Bochner and Lichnerowicz vanishing theorems, Weitzenbock formulas, and Finslerian spinors and Dirac operators. The tools developed in this book will find uses in several areas of physics and engineering, but especially in the mechanics of inhomogeneous media, e.g. Cofferat continua. Audience: This text will be of use to workers in stochastic processes, differential geometry, nonlinear analysis, epidemiology, ecology and evolution, as well as physics of the solid state and continua."

Nearly a hundred years have passed since Viggo Brun invented his famous sieve, and the use of sieve methods is constantly evolving. As probability and combinatorics have penetrated the fabric of mathematical activity, sieve methods have become more versatile and sophisticated and in recent years have played a part in some of the most spectacular mathematical discoveries. Many arithmetical investigations encounter a combinatorial problem that requires a sieving argument, and this tract offers a modern and reliable guide in such situations. The theory of higher dimensional sieves is thoroughly explored, and examples are provided throughout. A Mathematica (R) software package for sieve-theoretical calculations is provided on the authors' website. To further benefit readers, the Appendix describes methods for computing sieve functions. These methods are generally applicable to the computation of other functions used in analytic number theory. The appendix also illustrates features of Mathematica (R) which aid in the computation of such functions.

A pro-p group is the inverse limit of some system of finite p-groups, that is, of groups of prime-power order where the prime - conventionally denoted p - is fixed. Thus from one point of view, to study a pro-p group is the same as studying an infinite family of finite groups; but a pro-p group is also a compact topological group, and the compactness works its usual magic to bring 'infinite' problems down to manageable proportions. The p-adic integers appeared about a century ago, but the systematic study of pro-p groups in general is a fairly recent development. Although much has been dis covered, many avenues remain to be explored; the purpose of this book is to present a coherent account of the considerable achievements of the last several years, and to point the way forward. Thus our aim is both to stimulate research and to provide the comprehensive background on which that research must be based. The chapters cover a wide range. In order to ensure the most authoritative account, we have arranged for each chapter to be written by a leading contributor (or contributors) to the topic in question. Pro-p groups appear in several different, though sometimes overlapping, contexts."

Explores relationship between Fourier Analysis, convex geometry, and related areas; in the past, study of this relationship has led to important mathematical advances Presents new results and applications to diverse fields such as geometry, number theory, and analysis Contributors are leading experts in their respective fields Will be of interest to both pure and applied mathematicians

Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory 41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group theoretical principle, admitting an entirely elementary description. This de scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here."

This book is divided into two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. There are two principal topics: classification of quadratic forms and quadratic Diophantine equations. The second topic is a new framework which contains the investigation of Gauss on the sums of three squares as a special case. To make the book concise, the author proves some basic theorems in number theory only in some special cases. However, the book is self-contained when the base field is the rational number field, and the main theorems are stated with an arbitrary number field as the base field. So the reader familiar with class field theory will be able to learn the arithmetic theory of quadratic forms with no further references.

For a long time - at least from Fermat to Minkowski - the theory of quadratic forms was a part of number theory. Much of the best work of the great number theorists of the eighteenth and nineteenth century was concerned with problems about quadratic forms. On the basis of their work, Minkowski, Siegel, Hasse, Eichler and many others crea ted the impressive "arithmetic" theory of quadratic forms, which has been the object of the well-known books by Bachmann (1898/1923), Eichler (1952), and O'Meara (1963). Parallel to this development the ideas of abstract algebra and abstract linear algebra introduced by Dedekind, Frobenius, E. Noether and Artin led to today's structural mathematics with its emphasis on classification problems and general structure theorems. On the basis of both - the number theory of quadratic forms and the ideas of modern algebra - Witt opened, in 1937, a new chapter in the theory of quadratic forms. His most fruitful idea was to consider not single "individual" quadratic forms but rather the entity of all forms over a fixed ground field and to construct from this an algebra ic object. This object - the Witt ring - then became the principal object of the entire theory. Thirty years later Pfister demonstrated the significance of this approach by his celebrated structure theorems."

"Numerical Semigroups" is the first monograph devoted exclusively to the development of the theory of numerical semigroups. This concise, self-contained text is accessible to first year graduate students, giving the full background needed for readers unfamiliar with the topic. Researchers will find the tools presented useful in producing examples and counterexamples in other fields such as algebraic geometry, number theory, and linear programming.

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

This book provides a comprehensive account of a key (and perhaps the most important) theory upon which the Taylor-Wiles proof of Fermat's last theorem is based. The book begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond. It contains a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology groups. The final chapter presents a proof of a non-abelian class number formula and includes several new results from the author. The book will be of interest to graduate students and researchers in number theory (including algebraic and analytic number theorists) and arithmetic algebraic geometry.

The arithmetic properties of modular forms and elliptic curves lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula. Three main steps are outlined: the first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. Finiteness results for big Selmer groups are then established. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture. As the first book on the subject, the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases.

This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C.L. Siegel's work on quadratic forms. Serving as an introduction to the subject, these notes may also provide stimulation for further research.

The plan of this book had its inception in a course of lectures on arithmetical functions given by me in the summer of 1964 at the Forschungsinstitut fUr Mathematik of the Swiss Federal Institute of Technology, Zurich, at the invitation of Professor Beno Eckmann. My Introduction to Analytic Number Theory has appeared in the meanwhile, and this book may be looked upon as a sequel. It presupposes only a modicum of acquaintance with analysis and number theory. The arithmetical functions considered here are those associated with the distribution of prime numbers, as well as the partition function and the divisor function. Some of the problems posed by their asymptotic behaviour form the theme. They afford a glimpse of the variety of analytical methods used in the theory, and of the variety of problems that await solution. I owe a debt of gratitude to Professor Carl Ludwig Siegel, who has read the book in manuscript and given me the benefit of his criticism. I have improved the text in several places in response to his comments. I must thank Professor Raghavan Narasimhan for many stimulating discussions, and Mr. Henri Joris for the valuable assistance he has given me in checking the manuscript and correcting the proofs. K. Chandrasekharan July 1970 Contents Chapter I The prime number theorem and Selberg's method 1. Selberg's fonnula . . . . . . 1 2. A variant of Selberg's formula 6 12 3. Wirsing's inequality . . . . . 17 4. The prime number theorem. ."