This book was first published in 1952. It is largely devoted to the object of proving the Vinogradov-Goldbach theorem: that every sufficiently large odd number is the sum of three primes. In the course of proving this, T. Estermann, formerly Professor of Mathematics at the University of London, supplies numerous theories and results on characters and primes in arithmetic progressions. The author also ensures that the proofs presented to the reader are both clear and remarkably concise. The volume at hand addresses the Riemann zeta function, primes in arithmetical progression, and the ways in which odd numbers can be represented as the sum of three primes. At the end of the book is an index and a seven-page section of theorems and formulae for reference. This volume is both interesting and accessible, and will appeal to all with an enthusiasm for mathematics and problem solving.

The primary intent of the book is to introduce an array of beautiful problems in a variety of subjects quickly, pithily and completely rigorously to graduate students and advanced undergraduates. The book takes a number of specific problems and solves them, the needed tools developed along the way in the context of the particular problems. It treats a melange of topics from combinatorial probability theory, number theory, random graph theory and combinatorics. The problems in this book involve the asymptotic analysis of a discrete construct as some natural parameter of the system tends to infinity. Besides bridging discrete mathematics and mathematical analysis, the book makes a modest attempt at bridging disciplines. The problems were selected with an eye toward accessibility to a wide audience, including advanced undergraduate students. The book could be used for a seminar course in which students present the lectures."

In the spirit of Alladi Ramakrishnan's profound interest and contributions to three fields of science - Mathematics, Statistics, and Physics - this volume contains invited surveys and research articles from prominent members of these communities who also knew Ramakrishnan personally and greatly respected his influence in these areas of science. Historical photos, telegrams, and biographical narratives of Alladi Ramakrishnan's illustrious career of special interest are included as well.

Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms. As a consequence, algorithmic number theory has gradually emerged as an important and distinct field with connections to computer science and cryptography as well as other areas of mathematics. This 2008 text provides a comprehensive introduction to algorithmic number theory for beginning graduate students, written by the leading experts in the field. It includes several articles that cover the essential topics in this area, and in addition, there are contributions pointing in broader directions, including cryptography, computational class field theory, zeta functions and L-series, discrete logarithm algorithms, and quantum computing.

Serge Lang (1927-2005) was one of the top mathematicians of our time. He was born in Paris in 1927, and moved with his family to California, where he graduated from Beverly Hills High School in 1943. He subsequently graduated from California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951 before holding faculty positions at the University of Chicago and Columbia University (1955-1971). At the time of his death he was professor emeritus of Mathematics at Yale University.
An excellent writer, Lang has made innumerable and invaluable contributions in diverse fields of mathematics. He was perhaps best known for his work in number theory and for his mathematics textbooks, including the influential "Algebra." He was also a member of the Bourbaki group.
He was honored with the Cole Prize by the American Mathematical Society as well as with the Prix Carriere by the French Academy of Sciences. These five volumes collect the majority of his research papers, which range over a variety of topics."

Serge Lang is one of the top mathematicians of our time. Being an excellent writer, Lang has made innumerable contributions in diverse fields in mathematics and they are invaluable. He was honored with the Cole Prize by the American Mathematical Society as well as with the Prix Carriere by the French Academy of Sciences. In these four volumes 83 of his research papers are collected. They range over a variety of topics and will be of interest to many readers.

The last one hundred years have seen many important achievements in the classical part of number theory. After the proof of the Prime Number Theorem in 1896, a quick development of analytical tools led to the invention of various new methods, like Brun's sieve method and the circle method of Hardy, Littlewood and Ramanujan; developments in topics such as prime and additive number theory, and the solution of Fermat s problem. Rational Number Theory in the 20th Century: From PNT to FLT offers a short survey of 20th century developments in classical number theory, documenting between the proof of the Prime Number Theorem and the proof of Fermat's Last Theorem. The focus lays upon the part of number theory that deals with properties of integers and rational numbers. Chapters are divided into five time periods, which are then further divided into subject areas. With the introduction of each new topic, developments are followed through to the present day. This book will appeal to graduate researchers and student in number theory, however the presentation of main results without technicalities will make this accessible to anyone with an interest in the area."

Springer-Verlag has invited me to bring out my Selected Works. Being aware that Springer-Verlag enjoys high esteem in the scientific world as a reputed publisher, I have willingly accepted the offer. Immediately, I was faced with two problems. The first was that of acquaint ing the reader with the important stages in my scientific aetivities. For this purpose, I have included in the Selected Works eertain of my early works that have greatly influeneed my later studies. For the same reason, I have also in cluded in the book those works that contain the first, erude versions of the proofs for many of my basic theorems. The second problem was that of giving the reader the best possible opportunity to familiarize himself with the most important results and to learn to use my method. For this reason I have included the later improved versions of the proofs for my basic results, as weil as the monographs The Method of Trigo nometric Sums in Number Theory (Seeond Edition) and Special Variants of the Method of Trigonometric Sums."

This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of GL(n), and presents their associated L-functions. In Volume 1, the theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. In Volume 2, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises by Xander Faber, this book will motivate students and researchers to begin working in this fertile field of research.

Volume II is the second part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid, this 3-volume book aims to promote more deep understanding of ancient conception of the 'Universe Harmony,' the main conception of ancient Greek science, and implementation of this conception to modern science and education.This 3-volume book is a result of the authors' research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the 'Mathematics of Harmony,' a new interdisciplinary direction of modern science. This direction has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the generalized Binet's formulas), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational bases, Fibonacci computers, ternary mirror-symmetrical arithmetic).The books are intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science.

Volume III is the third part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid, this 3-volume book aims to promote more deep understanding of ancient conception of the 'Universe Harmony,' the main conception of ancient Greek science, and implementation of this conception to modern science and education.This 3-volume book is a result of the authors' research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the 'Mathematics of Harmony,' a new interdisciplinary direction of modern science. This direction has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the generalized Binet's formulas), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational bases, Fibonacci computers, ternary mirror-symmetrical arithmetic).The books are intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science.

Serge Lang (1927-2005) was one of the top mathematicians of our time. He was born in Paris in 1927, and moved with his family to California, where he graduated from Beverly Hills High School in 1943. He subsequently graduated from California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951 before holding faculty positions at the University of Chicago and Columbia University (1955-1971). At the time of his death he was professor emeritus of Mathematics at Yale University. An excellent writer, Lang has made innumerable and invaluable contributions in diverse fields of mathematics. He was perhaps best known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was also a member of the Bourbaki group. He was honored with the Cole Prize by the American Mathematical Society as well as with the Prix Carriere by the French Academy of Sciences. These five volumes collect the majority of his research papers, which range over a variety of topics.

Serge Lang (1927-2005) was one of the top mathematicians of our time. He was born in Paris in 1927, and moved with his family to California, where he graduated from Beverly Hills High School in 1943. He subsequently graduated from California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951 before holding faculty positions at the University of Chicago and Columbia University (1955-1971). At the time of his death he was professor emeritus of Mathematics at Yale University. An excellent writer, Lang has made innumerable and invaluable contributions in diverse fields of mathematics. He was perhaps best known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was also a member of the Bourbaki group. He was honored with the Cole Prize by the American Mathematical Society as well as with the Prix Carriere by the French Academy of Sciences. These five volumes collect the majority of his research papers, which range over a variety of topics"

These notes introduce a new class of algebraic curves on Hilbert modular surfaces. These curves are called twisted Teichmuller curves, because their construction is very reminiscent of Hirzebruch-Zagier cycles. These new objects are analyzed in detail and their main properties are described. In particular, the volume of twisted Teichmuller curves is calculated and their components are partially classified. The study of algebraic curves on Hilbert modular surfaces has been widely covered in the literature due to their arithmetic importance. Among these, twisted diagonals (Hirzebruch-Zagier cycles) are some of the most important examples.

Edmund Hlawka is a leading number theorist whose work has had a lasting influence on modern number theory and other branches of mathematics. He has contributed to diophantine approximation, the geometry of numbers, uniform distributions, analytic number theory, discrete geometry, convexity, numerical integration, inequalities, differential equations and gas dynamics. Of particular importance are his findings in the geometry of numbers (especially the Minkowski-Hlawka theorem) and uniform distribution. This Selecta volume collects his most important articles, many of which were previously hard to find. It will provide a useful tool for researchers and graduate students working in the areas covered, and includes a general introduction by E. Hlawka.

Previous publications on the generalization of the Thomae formulae to "Zn" curves have emphasized the theory's implications in mathematical physics and depended heavily on applied mathematical techniques. This book redevelops these previous results demonstrating how they can be derived directly from the basic properties of theta functions as functions on compact Riemann surfaces.

"Generalizations of Thomae's Formulafor "Zn" Curves" includes several refocused proofs developed in a generalized context that is more accessible to researchers in related mathematical fields such as algebraic geometry, complex analysis, and number theory.

This book is intended for mathematicians with an interest in complex analysis, algebraic geometry or number theory as well as physicists studying conformal field theory."

V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers.

Combinatorial research has proceeded vigorously in Russia over the last few decades, based on both translated Western sources and original Russian material. The present volume extends the extremal approach to the solution of a large class of problems, including some that were hitherto regarded as exclusively algorithmic, and broadens the choice of theoretical bases for modelling real phenomena in order to solve practical problems. Audience: Graduate students of mathematics and engineering interested in the thematics of extremal problems and in the field of combinatorics in general. Can be used both as a textbook and as a reference handbook.

The international symposium on number theory and analysis in memory of the late famous Chinese mathematician Prof. Hua Loo Keng was co-sponsored by the Institute of Mathematics, Academia Sinica and the University of Science and Technology of China. lt took place between August Ist and 7th of 1988 on the campus of Tsing Hua University, and some 150 mathematicians were pres- ent. The symposium was carried out in two separate sections: number theory and analysis. This is retlected in the publication ofa set oftwo volumes, the first one on Number Theory edited by Professor Wang Yuan and the second on Analysis by Professors Gong Sheng, Lu Qi-keng and Yang Lo. The distinguished list of main speakers and the contents of these two vol- umes reflect the high level of the mathematical activity throughout the seven days. W e pay special tribute to our main speakers professors Chuang, Conn, Ding, Drasin, Fitzgerald, Gaier, Gong, Grauert, Gu, Hejhal, Iyanaga, Karatsuba, Koranyi, Liao, Lu, Pan, Richert, Satake, Schmidt, Siu, Tatuzawa, Tsang, Vladimirov, Y. Wang, G. Y. Wang, Wustholz and Yang, who gave the excellent one hour lectures, and also to the participants who gave contributed talks on their own research work. The discussions among the mathematicians were always in a warm atmosphere. Our thanks go to professors Chern, Subbarao and Yau for their contributions to these proceedings.

In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the authors take an alternate approach to these theorems and generalize them to the setting of Hilbert modular varieties of arbitrary dimension. The approach is conceptual and uses tools that were not available to Hirzebruch and Zagier, including intersection homology theory, properties of modular cycles, and base change. Automorphic vector bundles, Hecke operators and Fourier coefficients of modular forms are presented both in the classical and adelic settings. The book should provide a foundation for approaching similar questions for other locally symmetric spaces."

This is an updated English translation of Cohomologie Galoisienne, published more than thirty years ago as one of the very first versions of Lecture Notes in Mathematics. It includes a reproduction of an influential paper by R. Steinberg, together with some new material and an expanded bibliography.

In recent years there has been an increasing interest in problems involving closed form evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ((s), the Hurwitz Zeta function ((s, a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(z, s, a). Some of these developments have apparently stemmed from an over two-century-old theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), from recent rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions."

The study of systems of special partial differential operators that arise naturally from the use of Clifford algebra as a calculus tool lies in the heart of Clifford analysis. The focus is on the study of Dirac operators and related ones, together with applications in mathematics, physics and engineering. At the present time, the study of Clifford algebra and Clifford analysis has grown into a major research field. There are two sources of papers in this collection. One is from a satellite conference to the ICM 2002 in Beijing, held August 15-18 at the University of Macau; and the other stems from invited contributions by top-notch experts in the field.