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Book review

Author(s) Korobov, N.M.
Title Exponential Sums and their Applications
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by V. Mihesan

The method of exponential sums is one of the few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers and its applications. The strongest results have been obtained with the aid of this method. Therefore knowledge of the fundamentals of the theory of exponential sums is necessary for studying modern number theory.

The main aim of this monograph is to present an as simple as possible exposition of the fundamentals of the theory and, with a series of examples, to show how exponential sums arise and are applied in problems of number theory and its applications.

The book is divided into three chapiters. It contains the classical results of Gauss and the methods of Weyl, Mordell and Vinogradov which are exposed in detail; the traditional applications of exponential sums to the distribution of fractional parts, the estimation of the Riemann zeta-function, the theory of congruences and Diophantine equations are considered too. Some new applications of exponential sums are also included.

The material is well written, and aesthetically presented, combining precision in statements and treatment with comments, examples, applications.

The book is intended for those who are beginning a study of exponential sums. At the same time, is can be interesting for specialists also, because it contrains some results which are not included in other monographs. Students and researches of number theory and numerical analysis are advised to buy it.