Exponential Sums and Their Applications

✍ Scribed by N. M. Korobov

• Publisher: Kluwer Academic Publishers
• Year: 1992
• Tongue: English
• Leaves: 225
• 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

Contents......Page all_6027_to_00224.cpc0006.djvu
Series Editor's Preface......Page all_6027_to_00224.cpc0004.djvu
Preface......Page all_6027_to_00224.cpc0008.djvu
Introduction......Page all_6027_to_00224.cpc0010.djvu
§1. Sums of the first degree......Page all_6027_to_00224.cpc0016.djvu
§2. General properties of complete sums......Page all_6027_to_00224.cpc0022.djvu
§3. Gaussian sums......Page all_6027_to_00224.cpc0028.djvu
§4. Simplest complete sums......Page all_6027_to_00224.cpc0037.djvu
§5. Mordell's method......Page all_6027_to_00224.cpc0044.djvu
§6. Systems of congruences......Page all_6027_to_00224.cpc0049.djvu
§7. Sums with exponential function......Page all_6027_to_00224.cpc0055.djvu
§8. Distribution of digits in complete period of periodic fractions......Page all_6027_to_00224.cpc0060.djvu
§9. Exponential sums with recurrent function......Page all_6027_to_00224.cpc0068.djvu
§10. Sums of Legendre's symbols......Page all_6027_to_00224.cpc0076.djvu
§11. Weyl's method......Page all_6027_to_00224.cpc0083.djvu
§12. Systems of equations......Page all_6027_to_00224.cpc0093.djvu
§13. Vinogradov's mean value theorem......Page all_6027_to_00224.cpc0102.djvu
§14. Estimates of Weyl's sums......Page all_6027_to_00224.cpc0112.djvu
§15. Repeated application of the mean value theorem......Page all_6027_to_00224.cpc0125.djvu
§16. Sums arising in zeta-function theory......Page all_6027_to_00224.cpc0134.djvu
§17. Incomplete rational sums......Page all_6027_to_00224.cpc0141.djvu
§18. Double exponential sums......Page all_6027_to_00224.cpc0148.djvu
§19. Uniform distribution of fractional parts......Page all_6027_to_00224.cpc0154.djvu
§20. Uniform distribution of functions systems and completely uniform distribution......Page all_6027_to_00224.cpc0164.djvu
§21. Normal and conjunctly normal numbers......Page all_6027_to_00224.cpc0174.djvu
§22. Distribution of digits in period part of periodical fractions......Page all_6027_to_00224.cpc0181.djvu
§23. Connection between exponential sums, quadrature formulas and fractional parts distribution......Page all_6027_to_00224.cpc0191.djvu
§24. Quadrature and interpolation formulas with the number-theoretical nets......Page all_6027_to_00224.cpc0202.djvu
References......Page all_6027_to_00224.cpc0218.djvu
Subject Index......Page all_6027_to_00224.cpc0222.djvu
Index of Names......Page all_6027_to_00224.cpc0224.djvu