Area, Lattice Points, and Exponential Sums (London Mathematical Society Monographs New Series)

In analytic number theory many problems can be reduced to those involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method for estimating the Riemann zeta function. Huxley and his coworkers have taken this method and vastly extended and improved it. The powerful techniques presented here go considerably beyond older methods for estimating exponential sums such as van de Corput's method. The potential for the method is far from being exhausted, and there is considerable motivation for other researchers to try to master this subject. However, anyone currently trying to learn all of this material has the formidable task of wading through numerous papers in the literature. This book simplifies that task by presenting all of the relevant literature and a good part of the background in one package. The book will find its biggest readership among mathematics graduate students and academics with a research interest in analytic theory; specifically exponential sum methods. This volume is concerned with the application of exponential sum techniques to a variety of problems in number theory, in particular the Riemann Zeta Function and the problem of estimating the number of lattice points in regions.