Trigonometric Sums in Number Theory and Analysis (De Gruyter Expositions in Mathematics, 39)

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.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject.

Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Aix-Marseille Université, France
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair , Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann , Columbia University, New York, USA Markus J. Pflaum , University of Colorado, Boulder, USA Dierk Schleicher , Aix-Marseille Université, France Katrin Wendland , Trinity College Dublin, Dublin, Ireland Honorary Editor Victor P. Maslov , Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

vinogradov First Developed The Method Of Trigonometric Sums In The First Decades Of The Twentieth Century As A Way Of Solving Problems In Analytical Number Theory. The Authors Here Present A Systematic Account Of The Theory Of Multiple Trigonometric Sums, Using A Unified Approach To Derive Results Similar To Those Of Vinogradov, With The Understanding The Theory Of Multiple Trigonometric Sums Has Reached The Level Of Completion Of One-dimensional Sums. They Use Their Results To Solve Problems In Analytic Number Theory And To Investigate Trigonometric Integrals Common In Physics, Statistics And Analysis. They Also Present Purely Mathematical Results For The Solvability Of Equations In Integers. This Text Is Designed For Graduate Students And Researchers In Number Theory, Probability Theory, And Analysis. Annotation ©2004 Book News, Inc., Portland, Or

TeX output 2004.09.22:1230 Preface 6 Basic Notation 8 Contents 10 Introduction 12 Chapter 1 Trigonometric integrals 17 Chapter 2 Rational trigonometric sums 59 Chapter 3 Weyl sums 90 Chapter 4 The mean value theorem for the multiple trigonometric sum 155 Chapter 5 Estimates for multiple trigonometric sums 192 Chapter 6 Several applications 221 Chapter 7 Special cases of the theory of multiple trigonometric sums 246 Chapter 8 The Hilbert–Kamke problem and its generalizations 327 Chapter 9 The p-adic method in three problems of number theory 372 Chapter 10 Estimates of multiple trigonometric sums with prime numbers 423 Chapter 11 Some applications of trigonometric sums and integrals 480 Chapter 12 Short Kloosterman sums 493 Appendix 547 Bibliography 550 Index 564 "In this book a systematic account of the theory of multiple trigonometric sums is given as created by the authors over a period of more than twenty years. The authors develop a unified approach with which they obtain estimates for these sums similar to the classical ones of I. M. Vinogradov. They use them to solve several problems in analytic number theory and investigate trigonometric integrals, which are often encountered in physics, mathematical statistics, and analysis. Moreover, purely arithmetical results concerning the solvability of equations in integers are presented." "This book is intended for graduate students and researchers in number theory, probability theory, and analysis." -- BOOK JACKET Das Buch stellt die von den Autoren entwickelte Theorie der mehrfachen trigonometrischen Summen vor. Einem integrierten Ansatz folgend, erhalten die Autoren Abschätzungen für diese Summen, vergleichbar den klassischen Abschätzungen von I. M. Vinogradov, und verwenden diese, um verschiedene Probleme der analytischen Zahlentheorie zu lösen. Sie untersuchen trigonometrische Integrale, die häufig in der Physik, der mathematischen Statistik und der Analysis anzutreffen sind, und präsentieren rein arithmetische Ergebnisse, die die Lösbarkeit von Gleichungen in ganzen Zahlen betreffen