Exponential Sums and Their Applications

✍ Scribed by N. M. Korobov

• Publisher: Kluwer Academic Publishers
• Year: 1992
• Tongue: English
• Leaves: 224
• Edition: Softcover reprint of hardcover 1st ed. 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

Exponential Sums and their Applications......Page 2
CONTENTS......Page 6
INTRODUCTION......Page 10
1. Sums of the first degree......Page 16
2. General properties of complete sums......Page 22
3. Gaussian sums......Page 28
4. Simplest complete sums......Page 37
5. Mordell's method......Page 44
6. Syste ms of congruences......Page 49
7. Sums with exponential function......Page 55
8. Distribution of digits in complete period of periodic fractions......Page 60
9. Exponential sums with recurrent fu nction......Page 68
10. Sums of Legendre's symbols......Page 76
11. Weyl's method......Page 83
12. Systems of equations......Page 93
13. Vinogradov's mean value theorem......Page 102
14. Estimates of Weyl's sums......Page 112
15. Repeat ed application of the mean value theorem......Page 125
16. Sums arising in zeta-function theory......Page 134
17. Incomplete rational......Page 141
18. Double exponential......Page 148
19. Uniform distribution of fr actional parts......Page 154
20. Uniform distribution of fu nctions systems and completely uniform distribution......Page 164
21. Normal and conjunctly normal numbers......Page 174
22. Distribution of digits in period part of periodical fractions......Page 181
23. Connection between exponential sums, quadrat ure formulas and fr actional parts distribution......Page 191
24. Quadrature and interpolation fo rmulas with the number-theoretical nets......Page 202
REFERENCES......Page 218
SUBJECT INDEX......Page 222
INDEX OF NAMES......Page 224