The Method of Trigonometrical Sums in the Theory of Numbers

Author: I. M. Vinogradov

Publisher: Courier Corporation

ISBN: 0486154521

Category: Mathematics

Page: 192

View: 5779

This text investigates Waring's problem, approximation by fractional parts of the values of a polynomial, estimates for Weyl sums, distribution of fractional parts of polynomial values, Goldbach's problem, more. 1954 edition.

Trigonometric Sums in Number Theory and Analysis

Author: Gennady I. Arkhipov,Vladimir N. Chubarikov,Anatoly A. Karatsuba

Publisher: Walter de Gruyter

ISBN: 3110197987

Category: Mathematics

Page: 564

View: 3337

The book presents the theory of multiple trigonometric sums constructed by the authors. Following a unified approach, the authors obtain estimates for these sums similar to the classical I. M. Vinogradov ́s estimates and use them to solve several problems in analytic number theory. They investigate trigonometric integrals, which are often encountered in physics, mathematical statistics, and analysis, and in addition they present purely arithmetic results concerning the solvability of equations in integers.

Complex Analysis in Number Theory

Author: Anatoly A. Karatsuba

Publisher: CRC Press

ISBN: 9780849328664

Category: Mathematics

Page: 208

View: 7339

This book examines the application of complex analysis methods to the theory of prime numbers. In an easy to understand manner, a connection is established between arithmetic problems and those of zero distribution for special functions. Main achievements in this field of mathematics are described. Indicated is a connection between the famous Riemann zeta-function and the structure of the universe, information theory, and quantum mechanics. The theory of Riemann zeta-function and, specifically, distribution of its zeros are presented in a concise and comprehensive way. The full proofs of some modern theorems are given. Significant methods of the analysis are also demonstrated as applied to fundamental problems of number theory.

Additive Theory of Prime Numbers

Author: Luogeng Hua

Publisher: American Mathematical Soc.

ISBN: 0821849425

Category: Mathematics

Page: 190

View: 1502

Loo-Keng Hua was a master mathematician, best known for his work using analytic methods in number theory. In particular, Hua is remembered for his contributions to Waring's Problem and his estimates of trigonometric sums. Additive Theory of Prime Numbers is an exposition of the classic methods as well as Hua's own techniques, many of which have now also become classic. An essential starting point is Vinogradov's mean-value theorem for trigonometric sums, which Hua usefully rephrases and improves. Hua states a generalized version of the Waring-Goldbach problem and gives asymptotic formulas for the number of solutions in Waring's Problem when the monomial $x^k$ is replaced by an arbitrary polynomial of degree $k$. The book is an excellent entry point for readers interested in additive number theory. It will also be of value to those interested in the development of the now classic methods of the subject.

Analytic Number Theory, Mathematical Analysis and Their Applications

Collection of Papers Dedicated to Academician Ivan Matveevich Vinogradov on His Ninetieth Birthday

Author: Nikolaĭ Nikolaevich Bogoli︠u︡bov,K. K. Mardzhanishvili

Publisher: American Mathematical Soc.

ISBN: 9780821830772

Category: Mathematics

Page: 247

View: 859

This Proceedings of the Steklov Institute of Mathematics, together with the volume preceding it (Volume 157), is a collection of papers dedicated to Academician I. M. Vinogradov on his ninetieth birthday. This volume contains original papers on various branches of mathematics: analytic number theory, algebra, partial differential equations, probability theory, and differential games.

Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis

Author: Hugh L. Montgomery

Publisher: American Mathematical Soc.

ISBN: 9780821889282

Category: Mathematics

Page: 220

View: 7948

This book contains lectures presented by Hugh L. Montgomery at the NSF-CBMS Regional Conference held at Kansas State University in May 1990. The book focuses on important topics in analytic number theory that involve ideas from harmonic analysis. One valuable aspect of the book is that it collects material that was either unpublished or that had appeared only in the research literature. This book would be an excellent resource for harmonic analysts interested in moving into research in analytic number theory. In addition, it is suitable as a textbook in an advanced graduate topics course in number theory.

Basic Analytic Number Theory

Author: Anatolij A. Karatsuba

Publisher: Springer Science & Business Media

ISBN: 3642580181

Category: Mathematics

Page: 222

View: 3135

This English translation of Karatsuba's Basic Analytic Number Theory follows closely the second Russian edition, published in Moscow in 1983. For the English edition, the author has considerably rewritten Chapter I, and has corrected various typographical and other minor errors throughout the the text. August, 1991 Melvyn B. Nathanson Introduction to the English Edition It gives me great pleasure that Springer-Verlag is publishing an English trans lation of my book. In the Soviet Union, the primary purpose of this monograph was to introduce mathematicians to the basic results and methods of analytic number theory, but the book has also been increasingly used as a textbook by graduate students in many different fields of mathematics. I hope that the English edition will be used in the same ways. I express my deep gratitude to Professor Melvyn B. Nathanson for his excellent translation and for much assistance in correcting errors in the original text. A.A. Karatsuba Introduction to the Second Russian Edition Number theory is the study of the properties of the integers. Analytic number theory is that part of number theory in which, besides purely number theoretic arguments, the methods of mathematical analysis play an essential role.

The Development of Prime Number Theory

From Euclid to Hardy and Littlewood

Author: Wladyslaw Narkiewicz

Publisher: Springer Science & Business Media

ISBN: 9783540662891

Category: Mathematics

Page: 449

View: 3877

1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's Elemen ta, where we find a proof of their infinitude, now regarded as canonical. One feels that Euclid's argument has its place in The Book, often quoted by the late Paul ErdOs, where the ultimate forms of mathematical arguments are preserved. Proofs of most other results on prime number distribution seem to be still far away from their optimal form and the aim of this book is to present the development of methods with which such problems were attacked in the course of time. This is not a historical book since we refrain from giving biographical details of the people who have played a role in this development and we do not discuss the questions concerning why each particular person became in terested in primes, because, usually, exact answers to them are impossible to obtain. Our idea is to present the development of the theory of the distribu tion of prime numbers in the period starting in antiquity and concluding at the end of the first decade of the 20th century. We shall also present some later developments, mostly in short comments, although the reader will find certain exceptions to that rule. The period of the last 80 years was full of new ideas (we mention only the applications of trigonometrical sums or the advent of various sieve methods) and certainly demands a separate book.

Number Theory, Algebra, Mathematical Analysis, and Their Applications

Dedicated to the 100th Anniversary of the Birth of Ivan Matveevich Vinogradov : Collection of Papers

Author: Ivan Matveevič Vinogradov (Mathematiker)

Publisher: American Mathematical Soc.

ISBN: 9780821831502

Category: Mathematics

Page: 373

View: 1717

This work is dedicated to the 100th anniversary of the birth of I. M. Vinogradov. It contains papers ranging over various areas of mathematics: including number theory; algebra; theory of functions of a real variable and of a complex variable; ordinary differential equations; optimal control; partial differential equations; mathematical physics; mechanics, and probability.

Introduction to Analytic Number Theory

Author: A. G. Postnikov

Publisher: American Mathematical Soc.

ISBN: 0821813498

Category: Mathematics

Page: 320

View: 7899

Aimed at a level between textbooks and the latest research monographs, this book is directed at researchers, teachers, and graduate students interested in number theory and its connections with other branches of science. Choosing to emphasize topics not sufficiently covered in the literature, the author has attempted to give as broad a picture as possible of the problems of analytic number theory.

Number Theory

Tradition and Modernization

Author: Wenpeng Zhang,Yoshio Tanigawa

Publisher: Springer Science & Business Media

ISBN: 0387308296

Category: Mathematics

Page: 234

View: 8442

This book collects survey and research papers on various topics in number theory. Although the topics and descriptive details appear varied, they are unified by two underlying principles: first, readability, and second, a smooth transition from traditional approaches to modern ones. Thus, on one hand, the traditional approach is presented in great detail, and on the other, the modernization of the methods in number theory is elaborated.

Sieves in Number Theory

Author: George Greaves

Publisher: Springer Science & Business Media

ISBN: 9783540416470

Category: Mathematics

Page: 304

View: 3587

Slightly more than 25 years ago, the first text devoted entirely to sieve meth ods made its appearance, rapidly to become a standard source and reference in the subject. The book of H. Halberstam and H.-E. Richert had actually been conceived in the mid-1960's. The initial stimulus had been provided by the paper of W. B. Jurkat and Richert, which determined the sifting limit for the linear sieve, using a combination of the ,A2 method of A. Selberg with combinatorial ideas which were in themselves of great importance and in terest. One of the declared objectives in writing their book was to place on record the sharpest form of what they called Selberg sieve theory available at the time. At the same time combinatorial methods were not neglected, and Halber stam and Richert included an account of the purely combinatorial method of Brun, which they believed to be worthy of further examination. Necessar ily they included only a briefer mention of the development of these ideas due (independently) to J. B. Rosser and H. Iwaniec that became generally available around 1971. These combinatorial notions have played a central part in subsequent developments, most notably in the papers of Iwaniec, and a further account may be thought to be timely. There have also been some developments in the theory of the so-called sieve with weights, and an account of these will also be included in this book.

Number Theory and Analysis

Proceedings of the International Conference on Number Theory, Celebrating the One Hundreth Anniversary of the Birth of Academician I.M. Vinogradov : Collection of Papers

Author: V. I. Blagodatskikh

Publisher: American Mathematical Soc.

ISBN: 9780821804674

Category: Mathematical analysis

Page: 351

View: 8668

This collection consists of papers by participants in the international conference celebrating the 100th anniversary of the birth of I. M. Vinogradov, held in Moscow in September 1991. The papers are devoted to current trends in analytic number theory, function theory, algebraic number theory, algebraic geometry, and combinatorics. The collection begins with a paper by A. A. Karatsuba that describes the life and scientific activity of Vinogradov.

Number Theory

Author: Jean-Marie De Koninck,Claude Levesque

Publisher: Walter de Gruyter

ISBN: 9783110117912

Category: Number theory

Page: 1002

View: 2946

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

The Rise and Development of the Theory of Series up to the Early 1820s

Author: Giovanni Ferraro

Publisher: Springer Science & Business Media

ISBN: 9780387734682

Category: Mathematics

Page: 392

View: 2718

The manuscript gives a coherent and detailed account of the theory of series in the eighteenth and early nineteenth centuries. It provides in one place an account of many results that are generally to be found - if at all - scattered throughout the historical and textbook literature. It presents the subject from the viewpoint of the mathematicians of the period, and is careful to distinguish earlier conceptions from ones that prevail today.