Van der Corput's Method of Exponential Sums

Abstract

This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums. These arise in many problems in analytic number theory. It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory. The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function. the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem.

Primitive lattice points in a thin strip along the boundary of a large convex planar domain

“…According to Krätzel [8], p. 57, (1/9, 13/18) is an exponent pair. By formula (3.3.4) in Graham & Kolesnik [2],…”

“…Vinogradov.) Graham & Kolesnik [2], Lemma 3.6, avoided this restriction but unfortunately produced an error term O( T /L) (in our notation) which is too crude for the present purpose. However, it is easy to construct what we need from the ideas in Graham & Kolesnik [2].…”

“…Graham & Kolesnik [2], Lemma 3.6, avoided this restriction but unfortunately produced an error term O( T /L) (in our notation) which is too crude for the present purpose. However, it is easy to construct what we need from the ideas in Graham & Kolesnik [2]. The Lemma in its present form was verified by the second named author in 1996 and first published with a proof in Kühleitner [9] (with permission) and also in Kühleitner & Nowak [10].…”

A mean value theorem for exponential sums

“…The sum over distinct m and n of l/(/n -n) 2 is O(M). The second set of integrated terms on the right of (4) has the same order of magnitude.…”

On the error term of a lattice counting problem

“…The argument is similar in the case ≥ 2k. which in turn is derived from the exponent pair (1.6) of Bourgain [4], where A and B denote applications of A-and B-processes, see [7].…”