Exponential Sums and their Applications (Mathematics and its Applications, 80)

✍ Scribed by N.M Korobov

• Publisher: Springer
• Year: 1992
• Tongue: English
• Leaves: 226
• Edition: 1992
• Category: Library

✦ Synopsis

The method of exponential sums is a general method enabling the solution of a wide range of problems in the theory of numbers and its applications. This volume presents an exposition of the fundamentals of the theory with the help of examples which show how exponential sums arise and how they are applied in problems of number theory and its applications.
The material is divided into three chapters which embrace the classical results of Gauss, and the methods of Weyl, Mordell and Vinogradov; the traditional applications of exponential sums to the distribution of fractional parts, the estimation of the Riemann zeta function; and the theory of congruences and Diophantine equations. Some new applications of exponential sums are also included.
It is assumed that the reader has a knowledge of the fundamentals of mathematical analysis and of elementary number theory.

✦ Table of Contents

Back cover
Front cover
Mathematics and Its Applications (Soviet Series)
Title page
Copyright page
SERIES EDITOR'S PREFACE
CONTENTS
PREFACE
INTRODUCTION
CHAPTER I. COMPLETE EXPONENTIAL SUMS
§1. Sums of the first degree
§2. General properties of complete sums
§3. Gaussian sums
§4. Simplest complete sums
§5. MordelPs method
§6. Systems of congruences
§7. Sums with exponential function
§8. Distribution of digits in complete period of periodic fractions
§9. Exponential sums with recurrent function
§10. Sums of Legendre's symbols
CHAPTER II. WEYL'S SUMS
§11. Weyl's method
§12. Systems of equations
§13. Vinogradov's mean value theorem
§14. Estimates of Weyl's sums
§15. Repeated application of the mean value theorem
§16. Sums arising in zeta-function theory
§17. Incomplete rational sums
§18. Double exponential sums
CHAPTER III. FRACTIONAL PARTS DISTRIBUTION, NORMAL NUMBERS, AND QUADRATURE FORMULAS
§19. Uniform distribution of fractional parts
§20. Uniform distribution of functions systems and completely uniform distribution
§21. Normal and conjunctly normal numbers
§22. Distribution of digits in period part of periodical fractions
§23. Connection between exponential sums, quadrature formulas and fractional parts distribution
§24. Quadrature and interpolation formulas with the number-theoretical nets
REFERENCES
SUBJECT INDEX
INDEX OF NAMES