# analytic number theory for undergraduates 3 monographs in number theory

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## Analytic Number Theory For Undergraduates

**Author :**Heng Huat Chan

**ISBN :**9789814365277

**Genre :**Mathematics

**File Size :**84. 14 MB

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This book is written for undergraduates who wish to learn some basic results in analytic number theory. It covers topics such as Bertrand's Postulate, the Prime Number Theorem and Dirichlet's Theorem of primes in arithmetic progression. The materials in this book are based on A Hildebrand's 1991 lectures delivered at the University of Illinois at Urbana-Champaign and the author's course conducted at the National University of Singapore from 2001 to 2008.

## Advanced Analytic Number Theory

**Author :**Carlos J. Moreno

**ISBN :**9780821842669

**Genre :**Mathematics

**File Size :**47. 34 MB

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Since the pioneering work of Euler, Dirichlet, and Riemann, the analytic properties of L-functions have been used to study the distribution of prime numbers. With the advent of the Langlands Program, L-functions have assumed a greater role in the study of the interplay between Diophantine questions about primes and representation theoretic properties of Galois representations. The present book provides a complete introduction to the most significant class of L-functions: the Artin-Hecke L-functions associated to finite-dimensional representations of Weil groups and to automorphic L-functions of principal type on the general linear group. In addition to establishing functional equations, growth estimates, and non-vanishing theorems, a thorough presentation of the explicit formulas of Riemann type in the context of Artin-Hecke and automorphic L-functions is also given. The survey is aimed at mathematicians and graduate students who want to learn about the modern analytic theory of L-functions and their applications in number theory and in the theory of automorphic representations. The requirements for a profitable study of this monograph are a knowledge of basic number theory and the rudiments of abstract harmonic analysis on locally compact abelian groups.

## Analytic Number Theory

**Author :**Paul T Bateman

**ISBN :**9789814365567

**Genre :**Mathematics

**File Size :**74. 10 MB

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' This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable (”elementary”) and complex variable (”analytic”) methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including transform and Tauberian methods, are developed as needed. Comments and corrigenda for the book are found at http://www.math.uiuc.edu/~diamond/. Contents:Calculus of Arithmetic FunctionsSummatory FunctionsThe Distribution of Prime NumbersAn Elementary Proof of the PNTDirichlet Series and Mellin TransformsInversion FormulasThe Riemann Zeta FunctionPrimes in Arithmetic ProgressionsApplications of CharactersOscillation TheoremsSievesApplication of SievesAppendix: Results from Analysis and Algebra Readership: Graduate students, academics and researchers interested in analytic number theory. Keywords:Analysis;Number TheoryReviews:“This book also includes a nice introduction to sieve methods … Overall, this is a nice well-written book with plenty of material for a one-year graduate course. It would also make nice supplementary reading for a student or researher learning the subject.”MAA Online Book Review “This is a nice introductory book on analytic number theory for students or readers with some background in real analysis, complex analysis, number theory and abstract algebra … There are various exercises throughout the entire book. Moreover, at the end of each chapter, historical backgrounds and developments of each particular subject or theorem are given together with references.”Mathematical Reviews “This book is suitable for beginning graduate students, or possibly even advanced undergraduates.”Zentralblatt MATH '

## A Primer Of Analytic Number Theory

**Author :**Jeffrey Stopple

**ISBN :**0521012538

**Genre :**Mathematics

**File Size :**66. 76 MB

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This 2003 undergraduate introduction to analytic number theory develops analytic skills in the course of studying ancient questions on polygonal numbers, perfect numbers and amicable pairs. The question of how the primes are distributed amongst all the integers is central in analytic number theory. This distribution is determined by the Riemann zeta function, and Riemann's work shows how it is connected to the zeroes of his function, and the significance of the Riemann Hypothesis. Starting from a traditional calculus course and assuming no complex analysis, the author develops the basic ideas of elementary number theory. The text is supplemented by series of exercises to further develop the concepts, and includes brief sketches of more advanced ideas, to present contemporary research problems at a level suitable for undergraduates. In addition to proofs, both rigorous and heuristic, the book includes extensive graphics and tables to make analytic concepts as concrete as possible.

## Analytic Number Theory

**Author :**Donald J. Newman

**ISBN :**9780387227405

**Genre :**Mathematics

**File Size :**55. 61 MB

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Some of the central topics in number theory, presnted in a simple and concise fashion. The author covers an amazing amount of material, despite a leisurely pace and emphasis on readability. His heartfelt enthusiasm enables readers to see what is magical about the subject. All the topics are presented in a refreshingly elegant and efficient manner with clever examples and interesting problems throughout. The text is suitable for a graduate course in analytic number theory.

## Analytic Number Theory

**Author :**Jean-Marie De Koninck

**ISBN :**9780821875773

**Genre :**Mathematics

**File Size :**63. 87 MB

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The authors assemble a fascinating collection of topics from analytic number theory that provides an introduction to the subject with a very clear and unique focus on the anatomy of integers, that is, on the study of the multiplicative structure of the integers. Some of the most important topics presented are the global and local behavior of arithmetic functions, an extensive study of smooth numbers, the Hardy-Ramanujan and Landau theorems, characters and the Dirichlet theorem, the $abc$ conjecture along with some of its applications, and sieve methods. The book concludes with a whole chapter on the index of composition of an integer. One of this book's best features is the collection of problems at the end of each chapter that have been chosen carefully to reinforce the material. The authors include solutions to the even-numbered problems, making this volume very appropriate for readers who want to test their understanding of the theory presented in the book.

## Number Theory Fermat S Dream

**Author :**Kazuya Kato

**ISBN :**082180863X

**Genre :**Mathematics

**File Size :**24. 81 MB

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This is the English translation of the original Japanese book. In this volume, ``Fermat's Dream'', core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series.

## Not Always Buried Deep

**Author :**Paul Pollack

**ISBN :**9780821848807

**Genre :**Mathematics

**File Size :**85. 8 MB

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Number theory is one of the few areas of mathematics where problems of substantial interest can be fully described to someone with minimal mathematical background. Solving such problems sometimes requires difficult and deep methods. But this is not a universal phenomenon; many engaging problems can be successfully attacked with little more than one's mathematical bare hands. In this case one says that the problem can be solved in an elementary way. Such elementary methods and the problems to which they apply are the subject of this book. Not Always Buried Deep is designed to be read and enjoyed by those who wish to explore elementary methods in modern number theory. The heart of the book is a thorough introduction to elementary prime number theory, including Dirichlet's theorem on primes in arithmetic progressions, the Brun sieve, and the Erdos-Selberg proof of the prime number theorem. Rather than trying to present a comprehensive treatise, Pollack focuses on topics that are particularly attractive and accessible. Other topics covered include Gauss's theory of cyclotomy and its applications to rational reciprocity laws, Hilbert's solution to Waring's problem, and modern work on perfect numbers. The nature of the material means that little is required in terms of prerequisites: The reader is expected to have prior familiarity with number theory at the level of an undergraduate course and a first course in modern algebra (covering groups, rings, and fields). The exposition is complemented by over 200 exercises and 400 references.

## A Course In Analytic Number Theory

**Author :**Marius Overholt

**ISBN :**9781470417062

**Genre :**Mathematics

**File Size :**82. 88 MB

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This book is an introduction to analytic number theory suitable for beginning graduate students. It covers everything one expects in a first course in this field, such as growth of arithmetic functions, existence of primes in arithmetic progressions, and the Prime Number Theorem. But it also covers more challenging topics that might be used in a second course, such as the Siegel-Walfisz theorem, functional equations of L-functions, and the explicit formula of von Mangoldt. For students with an interest in Diophantine analysis, there is a chapter on the Circle Method and Waring's Problem. Those with an interest in algebraic number theory may find the chapter on the analytic theory of number fields of interest, with proofs of the Dirichlet unit theorem, the analytic class number formula, the functional equation of the Dedekind zeta function, and the Prime Ideal Theorem. The exposition is both clear and precise, reflecting careful attention to the needs of the reader. The text includes extensive historical notes, which occur at the ends of the chapters. The exercises range from introductory problems and standard problems in analytic number theory to interesting original problems that will challenge the reader. The author has made an effort to provide clear explanations for the techniques of analysis used. No background in analysis beyond rigorous calculus and a first course in complex function theory is assumed.

## Introduction To The Theory Of Numbers

**Author :**Harold N. Shapiro

**ISBN :**9780486466699

**Genre :**Mathematics

**File Size :**65. 22 MB

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Starting with the fundamentals of number theory, this text advances to an intermediate level. Author Harold N. Shapiro, Professor Emeritus of Mathematics at New York University's Courant Institute, addresses this treatment toward advanced undergraduates and graduate students. Selected chapters, sections, and exercises are appropriate for undergraduate courses. The first five chapters focus on the basic material of number theory, employing special problems, some of which are of historical interest. Succeeding chapters explore evolutions from the notion of congruence, examine a variety of applications related to counting problems, and develop the roots of number theory. Two "do-it-yourself" chapters offer readers the chance to carry out small-scale mathematical investigations that involve material covered in previous chapters.