Trigonometric Sums in Number Theory and Analysis

The book presents the theory of multiple trigonometric sums constructed by the authors. Following a unified approach, the authors obtain estimates for these sums similar to the classical I. M. Vinogradov s estimates and use them to solve several problems in analytic number theory. They investigate trigonometric integrals, which are often encountered in physics, mathematical statistics, and analysis, and in addition they present purely arithmetic results concerning the solvability of equations in integers.

1;Frontmatter;1 2;Contents;9 3;Introduction;11 4;Chapter 1. Trigonometric integrals;16 5;Chapter 2. Rational trigonometric sums;58 6;Chapter 3. Weyl sums;89 7;Chapter 4. Mean value theorems for multiple trigonometric sums;154 8;Chapter 5. Estimates for multiple trigonometric sums;191 9;Chapter 6. Several applications;220 10;Chapter 7. Special cases of the theory of multiple trigonometric sums;245 11;Chapter 8. The HilbertKamke problem and its generalizations;326 12;Chapter 9. The p-adic method in three problems of number theory;371 13;Chapter 10. Estimates of multiple trigonometric sums with prime numbers;422 14;Chapter 11. Some applications of trigonometric sums and integrals;479 15;Chapter 12. Short Kloosterman sums;492 16;Backmatter;547