Exponential Sums and Their Applications

✍ Scribed by Nikolaĭ Mikhaĭlovich Korobov

• Publisher: Kluwer Academic Publishers
• Year: 1992
• Tongue: English
• Leaves: 227
• Series: Mathematics and Its Applications (Soviet Series) volume 80
• Edition: 1
• 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

Cover......Page 1
Series Editors......Page 2
Title: Exponential Sums and their Applications......Page 3
ISBN 0-7923·-1647-9......Page 4
Series Editor's Preface......Page 6
CONTENT'S......Page 8
PREFACE......Page 10
INTRODUCTION......Page 12
§ 1. Sums of the first degree......Page 18
§ 2. General properties of complete sums......Page 0
§ 3. Gaussian sums......Page 30
§ 4. Simplest complete sums......Page 39
§ 5. Mordell's method......Page 46
§ 6. Systems of congruences......Page 51
§ 7. Sums with exponential function......Page 57
§ 8. Distribution of digits in complete period of periodic fractions......Page 62
§ 9. Exponential sums with recurrent function......Page 70
§ 10. Sums of Legendre's symbols......Page 78
§ 11. Weyl's method......Page 85
§ 12. Systems of equations......Page 95
§ 13. Vlnogradov's mean value theorem......Page 104
§ 14. Estimates of Weyl's sums......Page 114
§ 15. Repeated applicatioll of the mean value theorem......Page 127
§ 16. Sums arising in zeta-function theory......Page 136
§ 17. Incomplete rational sums......Page 143
§ 18. Double exponential sums......Page 150
§ 19. Uniform distribution of fractional parts......Page 156
§ 20. Uni,form distribution of functions systems and completely uniform distribution......Page 166
§ 21. Normal and conjunctly normal numbers......Page 176
§ 22. Distribution of digits in period part of periodical fractions......Page 183
§ 23. Connection between exponential sums, quadrature formulas and fractional parts distribution......Page 193
§ 24. Quadrature and interpolation formulas with the number-theoretical nets......Page 204
REFERENCES......Page 220
SUBJ'ECT INDEX......Page 224
IN'DEX OF NAMES......Page 226
Back Cover......Page 227