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.

Dr. Gennady I. Arkhipov and Dr. Anatoly A. Karatsuba are working at the Steklov Mathematical Institute of the Russian Academy of Sciences at Moscow, Russia.

Vladimir N. Chubarikov is Professor at the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University, Russia.