This book is devoted to a special problem of number theory, that is the estimation of the number of lattice points in large closed domains of ordinary Euclidean spaces. This problem is closely related to the problem of estimation of exponential sums. Van der Corput developed an important method of dealing with such sums at the beginning of our century. His method has initiated a new period of intensive development during the last few years. Beside this method some interesting applications of this theory to the circle and sphere problems, to Dirichlet's divisor problem and some generalizations of these problems are considered in this book. The distribution of powerful numbers and finite Abelian groups are also investigated. It is, however, impossible to describe all the interesting problems. Furthermore, the estimations proved are not the very best in all cases. Further improvements can be obtained, however, by refinements of the method; whereupon many technical difficulties arise. The object of this book is to acquaint the reader with the fundamental results and methods so that he is able to follow up the original papers. I wish to express my gratitude to G. Horn, H. Menzer and L. Schnabel for reading the manuscript and for correcting a large number of mistakes. I should also like to this opportunity to thank M. Fritsch for valuable assistance in checking the English.
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