Exponential Sums and Their Applications

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. 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

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.