In the English edition, the chapter on the Geometry of Numbers has been enlarged to include the important findings of H. Lenstraj furthermore, tried and tested examples and exercises have been included. The translator, Prof. Charles Thomas, has solved the difficult problem of the German text into English in an admirable way. He deserves transferring our 'Unreserved praise and special thailks. Finally, we would like to express our gratitude to Springer-Verlag, for their commitment to the publication of this English edition, and for the special care taken in its production. Vienna, March 1991 E. Hlawka J. SchoiBengeier R. Taschner Preface to the German Edition We have set ourselves two aims with the present book on number theory. On the one hand for a reader who has studied elementary number theory, and who has knowledge of analytic geometry, differential and integral calculus, together with the elements of complex variable theory, we wish to introduce basic results from the areas of the geometry of numbers, diophantine ap proximation, prime number theory, and the asymptotic calculation of number theoretic functions. However on the other hand for the student who has al ready studied analytic number theory, we also present results and principles of proof, which until now have barely if at all appeared in text books.

This book is an attempt to describe the gradual development of the major schools of research on number theory in South India, Punjab, Mumbai, Bengal, and Bihar-including the establishment of Tata Institute of Fundamental Research (TIFR), Mumbai, a landmark event in the history of research of number theory in India. Research on number theory in India during modern times started with the advent of the iconic genius Srinivasa Ramanujan, inspiring mathematicians around the world. This book discusses the national and international impact of the research made by Indian number theorists. It also includes a carefully compiled, comprehensive bibliography of major 20th century Indian number theorists making this book important from the standpoint of historic documentation and a valuable resource for researchers of the field for their literature survey. This book also briefly discusses the importance of number theory in the modern world of mathematics, including applications of the results developed by indigenous number theorists in practical fields. Since the book is written from the viewpoint of the history of science, technical jargon and mathematical expressions have been avoided as much as possible.

Hypergeometric functions have occupied a significant position in mathematics for over two centuries. This monograph, by one of the foremost experts, is concerned with the Boyarsky principle which expresses the analytical properties of a certain proto-gamma function. Professor Dwork develops here a theory which is broad enough to encompass several of the most important hypergeometric functions in the literature and their cohomology. A central theme is the development of the Laplace transform in this context and its application to spaces of functions associated with hypergeometric functions. Consequently, this book represents a significant further development of the theory and demonstrates how the Boyarsky principle may be given a cohomological interpretation. The author includes an exposition of the relationship between this theory and Gauss sums and generalized Jacobi sums, and explores the theory of duality which throws new light on the theory of exponential sums and confluent hypergeometric functions.

The area of computational cryptography is dedicated to the development of effective methods in algorithmic number theory that improve implementation of cryptosystems or further their cryptanalysis. This book is a tribute to Arjen K. Lenstra, one of the key contributors to the field, on the occasion of his 65th birthday, covering his best-known scientific achievements in the field. Students and security engineers will appreciate this no-nonsense introduction to the hard mathematical problems used in cryptography and on which cybersecurity is built, as well as the overview of recent advances on how to solve these problems from both theoretical and practical applied perspectives. Beginning with polynomials, the book moves on to the celebrated Lenstra-Lenstra-Lovasz lattice reduction algorithm, and then progresses to integer factorization and the impact of these methods to the selection of strong cryptographic keys for usage in widely used standards.

This textbook covers a wide array of topics in analytic and multiplicative number theory, suitable for graduate level courses.Extensively revised and extended, this Advanced Edition takes a deeper dive into the subject, with the elementary topics of the previous edition making way for a fuller treatment of more advanced topics. The core themes of the distribution of prime numbers, arithmetic functions, lattice points, exponential sums and number fields now contain many more details and additional topics. In addition to covering a range of classical and standard results, some recent work on a variety of topics is discussed in the book, including arithmetic functions of several variables, bounded gaps between prime numbers a la Yitang Zhang, Mordell's method for exponential sums over finite fields, the resonance method for the Riemann zeta function, the Hooley divisor function, and many others. Throughout the book, the emphasis is on explicit results. Assuming only familiarity with elementary number theory and analysis at an undergraduate level, this textbook provides an accessible gateway to a rich and active area of number theory. With an abundance of new topics and 50% more exercises, all with solutions, it is now an even better guide for independent study.

This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.

Robert Langlands formulated his celebrated conjectures, initiating the Langlands Program, at the age of 31, profoundly changing the landscape of mathematics. Langlands, recipient of the Abel Prize, is famous for his insight in discovering links among seemingly dissimilar objects, leading to astounding results. This book is uniquely designed to serve a wide range of mathematicians and advanced students, showcasing Langlands' unique creativity and guiding readers through the areas of Langlands' work that are generally regarded as technical and difficult to penetrate. Part 1 features non-technical personal reflections, including Langlands' own words describing how and why he was led to formulate his conjectures. Part 2 includes survey articles of Langlands' early work that led to his conjectures, and centers on his principle of functoriality and foundational work on the Eisenstein series, and is accessible to mathematicians from other fields. Part 3 describes some of Langlands' contributions to mathematical physics.

This book deals with the development of Diophantine problems starting with Thue's path breaking result and culminating in Roth's theorem with applications. It discusses classical results including Hermite-Lindemann-Weierstrass theorem, Gelfond-Schneider theorem, Schmidt's subspace theorem and more. It also includes two theorems of Ramachandra which are not widely known and other interesting results derived on the values of Weierstrass elliptic function. Given the constantly growing number of applications of linear forms in logarithms, it is becoming increasingly important for any student wanting to work in this area to know the proofs of Baker's original results. This book presents Baker's original results in a format suitable for graduate students, with a focus on presenting the content in an accessible and simple manner. Each student-friendly chapter concludes with selected problems in the form of "Exercises" and interesting information presented as "Notes," intended to spark readers' curiosity.

Numbers and politics are inter-related at almost every level -- be it the abstract geometry of understandings of territory, the explosion of population statistics and measures of economic standards, the popularity of Utilitarianism, Rawlsian notions of justice, the notion of value, or simply the very idea of political science. Time and space are reduced to co-ordinates, illustrating a very real take on the political: a way of measuring and controlling it. This book engages with the relation between politics and number through a reading, exegesis and critique of the work of Martin Heidegger. The importance of mathematics and the role played by the understandings of calculation is a recurrent concern in his writing and is regularly contrasted with understandings of speech and language. This book provides the most detailed analysis of the relation between language, politics and mathematics in Heidegger's work. It insists that questions of language and calculation in Heidegger are inherently political, and that a far broader range of his work is concerned with politics than is usually admitted. Key Features: *A unique introduction to the political dimension of Heidegger's work, opening it up to a wider audience *Offers an original exploration of the relationship between language, mathematics and politics in Heidegger's thinking *Shows how questions of politics and calculation are inter-related in modern conceptions of the political Books in the series are...Valentine and Arditi Polemicization Shapiro Cinematic Political Thought Chambers Untimely Politics Elden Speaking Against Number Bowman Post-Marxism Versus Cultural Studies Marchart Post-Foundational Political Thought Little Democratic Piety

Work examines the latest algorithms and tools to solve classical types of diophantine equations.; Unique book---closest competitor, Smart, Cambridge, does not treat index form equations.; Author is a leading researcher in the field of computational algebraic number theory.; The text is illustrated with several tables of various number fields, including their data on power integral bases.; Several interesting properties of number fields are examined.; Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine equations.

This book takes the reader on a journey from familiar high school mathematics to undergraduate algebra and number theory. The journey starts with the basic idea that new number systems arise from solving different equations, leading to (abstract) algebra. Along this journey, the reader will be exposed to important ideas of mathematics, and will learn a little about how mathematics is really done.Starting at an elementary level, the book gradually eases the reader into the complexities of higher mathematics; in particular, the formal structure of mathematical writing (definitions, theorems and proofs) is introduced in simple terms. The book covers a range of topics, from the very foundations (numbers, set theory) to basic abstract algebra (groups, rings, fields), driven throughout by the need to understand concrete equations and problems, such as determining which numbers are sums of squares. Some topics usually reserved for a more advanced audience, such as Eisenstein integers or quadratic reciprocity, are lucidly presented in an accessible way. The book also introduces the reader to open source software for computations, to enhance understanding of the material and nurture basic programming skills. For the more adventurous, a number of Outlooks included in the text offer a glimpse of possible mathematical excursions. This book supports readers in transition from high school to university mathematics, and will also benefit university students keen to explore the beginnings of algebraic number theory. It can be read either on its own or as a supporting text for first courses in algebra or number theory, and can also be used for a topics course on Diophantine equations.

This volume introduces some recent developments in Arithmetic Geometry over local fields. Its seven chapters are centered around two common themes: the study of Drinfeld modules and non-Archimedean analytic geometry. The notes grew out of lectures held during the research program "Arithmetic and geometry of local and global fields" which took place at the Vietnam Institute of Advanced Study in Mathematics (VIASM) from June to August 2018. The authors, leading experts in the field, have put great effort into making the text as self-contained as possible, introducing the basic tools of the subject. The numerous concrete examples and suggested research problems will enable graduate students and young researchers to quickly reach the frontiers of this fascinating branch of mathematics.

These proceedings collect several number theory articles, most of which were written in connection to the workshop WIN4: Women in Numbers, held in August 2017, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. It collects papers disseminating research outcomes from collaborations initiated during the workshop as well as other original research contributions involving participants of the WIN workshops. The workshop and this volume are part of the WIN network, aimed at highlighting the research of women and gender minorities in number theory as well as increasing their participation and boosting their potential collaborations in number theory and related fields.

This volume presents a collection of results related to the BSD conjecture, based on the first two India-China conferences on this topic. It provides an overview of the conjecture and a few special cases where the conjecture is proved. The broad theme of the two conferences was "Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture". The first was held at Beijing International Centre for Mathematical Research (BICMR) in December 2014 and the second was held at the International Centre for Theoretical Sciences (ICTS), Bangalore, India in December 2016. Providing a broad overview of the subject, the book is a valuable resource for young researchers wishing to work in this area. The articles have an extensive list of references to enable diligent researchers to gain an idea of the current state of art on this conjecture.

In this book, the author pays tribute to Bernhard Riemann (1826-1866), a mathematician with revolutionary ideas, whose work on the theory of integration, the Fourier transform, the hypergeometric differential equation, etc. contributed immensely to mathematical physics. The text concentrates in particular on Riemann's only work on prime numbers, including ideas - new at the time - such as analytical continuation into the complex plane and the product formula for entire functions. A detailed analysis of the zeros of the Riemann zeta-function is presented. The impact of Riemann's ideas on regularizing infinite values in field theory is also emphasized. This revised and enhanced new edition contains three new chapters, two on the application of Riemann's zeta-function regularization to obtain the partition function of a Bose (Fermi) oscillator and one on the zeta-function regularization in quantum electrodynamics. Appendix A2 has been re-written to make the calculations more transparent. A summary of Euler-Riemann formulae completes the book.

Den beiden Autoren ist es auf hervorragende Weise gelungen, das Ziel ihres Buches zu verwirklichen und dem interessierten Nichtmathematiker einen tiefen Einblick in das Wesen der Mathematik, ihre SchAnheit und Tiefe zu ermAglichen. So werden in 26 in sich abgeschlossenen Kapiteln ausgewAhlte Themen der klassischen Mathematik - unter anderem Probleme der Zahlentheorie, der analytischen Geometrie und der Topologie - in fesselnder Form vorgetragen. Der Schwerpunkt liegt dabei aber nicht auf der mathematisch-stofflichen Tatsache, sondern auf dem Ablauf des Geschehens, auf der Methode der Fragestellung und auf der Methode, gestellte Fragen zu lAsen. Dieses Buch fA1/4r Liebhaber der Mathematik hat seit seinem Erscheinen im Jahre 1930 nichts von seiner Frische und Faszination verloren.

This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory.Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Cebotarev density theorem. Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.

The natural numbers have been studied for thousands of years, yet most undergraduate textbooks present number theory as a long list of theorems with little mention of how these results were discovered or why they are important. This book emphasizes the historical development of number theory, describing methods, theorems, and proofs in the contexts in which they originated, and providing an accessible introduction to one of the most fascinating subjects in mathematics. Written in an informal style by an award-winning teacher, Number Theory covers prime numbers, Fibonacci numbers, and a host of other essential topics in number theory, while also telling the stories of the great mathematicians behind these developments, including Euclid, Carl Friedrich Gauss, and Sophie Germain. This one-of-a-kind introductory textbook features an extensive set of problems that enable students to actively reinforce and extend their understanding of the material, as well as fully worked solutions for many of these problems. It also includes helpful hints for when students are unsure of how to get started on a given problem. * Uses a unique historical approach to teaching number theory * Features numerous problems, helpful hints, and fully worked solutions * Discusses fun topics like Pythagorean tuning in music, Sudoku puzzles, and arithmetic progressions of primes * Includes an introduction to Sage, an easy-to-learn yet powerful open-source mathematics software package * Ideal for undergraduate mathematics majors as well as non-math majors * Digital solutions manual (available only to professors)

This innovative undergraduate textbook approaches number theory through the lens of abstract algebra. Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory. Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics. The text is organized around three core themes: the notion of what a "number" is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems. Other aspects of modern number theory - including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings - are included in smaller but persistent threads woven through chapters and exercise sets. Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL "Exploration" worksheets appear in many sections, some of which involve numerical investigations. To assist students who may not have experience with programming languages, Python worksheets are available on the book's website. The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects. The topics covered in these explorations are public key cryptography, Lagrange's four-square theorem, units and Pell's Equation, various cases of the solution to Fermat's Last Theorem, and a peek into other deeper mysteries of algebraic number theory. Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the "numberverse."

This is the second in a series of three volumes dealing with important topics in algebra. Volume 2 is an introduction to linear algebra (including linear algebra over rings), Galois theory, representation theory, and the theory of group extensions. The section on linear algebra (chapters 1-5) does not require any background material from Algebra 1, except an understanding of set theory. Linear algebra is the most applicable branch of mathematics, and it is essential for students of science and engineering As such, the text can be used for one-semester courses for these students. The remaining part of the volume discusses Jordan and rational forms, general linear algebra (linear algebra over rings), Galois theory, representation theory (linear algebra over group algebras), and the theory of extension of groups follow linear algebra, and is suitable as a text for the second and third year students specializing in mathematics.

This is the first in a series of three volumes dealing with important topics in algebra. It offers an introduction to the foundations of mathematics together with the fundamental algebraic structures, namely groups, rings, fields, and arithmetic. Intended as a text for undergraduate and graduate students of mathematics, it discusses all major topics in algebra with numerous motivating illustrations and exercises to enable readers to acquire a good understanding of the basic algebraic structures, which they can then use to find the exact or the most realistic solutions to their problems.

This highly readable book aims to ease the many challenges of starting undergraduate research. It accomplishes this by presenting a diverse series of self-contained, accessible articles which include specific open problems and prepare the reader to tackle them with ample background material and references. Each article also contains a carefully selected bibliography for further reading. The content spans the breadth of mathematics, including many topics that are not normally addressed by the undergraduate curriculum (such as matroid theory, mathematical biology, and operations research), yet have few enough prerequisites that the interested student can start exploring them under the guidance of a faculty member. Whether trying to start an undergraduate thesis, embarking on a summer REU, or preparing for graduate school, this book is appropriate for a variety of students and the faculty who guide them.

Mumford is a well-known mathematician and winner of the Fields Medal, the highest honor available in mathematics Many of these papers are currently unavailable, and the correspondence with Grothendieck has never before been published